Algebra 2 v2

Perfect Square Tinomial (plus version) (a + b)²

Think of it like (a+b)(a+b) so... (a + b)² = a² +2ab + b²

substitution method

solve for one variable then plug in

linear equations

system that contains lines so no x² see example above

Write a quadratic function in standard form with zeros 2 and -1

1. Write the zeros as solutions for two equations: x = 2 or x = -1 2. Re-write each equation so that it equals 0: x - 2 = 0 or x + 1 = 0 3. Make a single equation, based on Zero Product Rule (anything times 0 is 0): (x - 2)(x + 1) = 0 (Because if either x - 2 = 0 or x + 1 = 0, the product of those two expressions will be 0 4. Put into standard form, by multiplying (x - 2)(x + 1) out: x² - x - 2 = f(x) If you graph this function, you'll see that it crosses the x axis (i.e. it has a zero value for f(x), at the original zeros of 2 and -1.

linear function

A linear function is a function whose graph is a straight line. The line can't be vertical, since then we wouldn't have a function, but any other sort of straight line is fine.

inconsistent system

A system of two linear equations can have one solution, an infinite number of solutions, or no solution. For a inconsistent system it would have no solutions. where the lines are parallel and don't intersect.

What is vertical stretch or compression

Changing the shape of the curve by compressing or stretching it out. f(x) = af(x). The a will stretch or compress. It will stretch if a is greater than 1, and will compress if a is less than 1.

What is the product property of logarithms?

For exponents, the product property says that bⁿbⁿ = bⁿ⁺ⁿ. For example, b³b⁵ = b³⁺⁵ = b⁸. When you multiply the base, you add the exponents. In logs, this works the same way -- log₃mn = log₃m + log₃n. When you are finding the log of a product, you can add the logs of the two factors of the product.

How to solve equation where unknown is in the exponent is can't convert it all to the same base

If can't convert the equation to put everything in the same base, take the log of both sides. if x = y, then logx = logy Example. 7ⁿ⁻³ = 350. (use base 10 just to make it easy) log7ⁿ⁻³ = log350 (n-3)log7 = log 350 (log7)x - 3log7 = log350 (log7)x = log350 +3log7 x = (log350 +3log7)/log7

What does a table of an exponential function look like

In exponential function, the y values go up not be a constant amount but by a constant ratio. So, for example, as x goes up by one, you always multiply the prior y value by the same number. So, as you go along the x axis, you are multiplying your base by itself one additional time for each increase of one unit of x. Say the base if 4 and the function is y = 4ⁿ, when n is 2 we are squaring 4. When n is 3 we are cubing 4. When n is 4 we are taking 4 to the fourth power. So, for every single digit increase in n, we multiple 4 by itself one more time. This leads to very steep curve because the numbers get very large very fast.

What is formula for natural decay

N(t) = N(0) e^-λ t Here N(t) is the quantity at time t, and N0 = N(0) is the initial quantity, i.e. the quantity at time t = 0. λ is the natural decay constant (which is a function of half life)

What is the Standard form for Quadratic Equation

Standard form is f(x) = ax² + bx + c

What is a way to figure out the inverse of a function by switching x and y.

Start with the function f(x) = 2x -5. That's the same as saying y = 2x - 5. Then just switch the x and y, and you get x = 2y - 5, and then solve for y. So you add 5 to both sides and get x + 5 = 2y. Then divide both sides by 2, and you get y = (x+5)/2

What does it mean to "find the zeros" of a quadratic function

The "zero" of a function is the value of the input x that makes the output f(x) (also known as y) equal zero. These are the x intercepts, where the function line cross the x axis (i.e where y = 0). They are the places in where y = 0. Linear functions can have no more than one zero. Quadratic functions can have two zeros because a parabola can cross the x axis at two different points.

What happens to domain and range of an inverse function

The domain (the possible x values) of the first function become the range (possible y values) of the inverse. And the range of the first function (the possible y values) becomes the domain of the inverse (the possible x values)

What is an asymtote

The line that an exponential function approaches, but never quite touches, is called the asymptote. For example, if f(n) = 1/2ⁿ, then as n gets higher, f(n) gets closer and closer to zero, but never touches zero. So, zero is the asymptote. ½² = ¼ and ½³ = 1/8 and ½⁴ = 1/32, etc.

What is the log chant

The log is just an exponent

What is a parent function?

The parent function is the basic function that involves the highest degree of the variable, or the log. Example, for the function f(x) = x² + 3, the parent function is f(x) = x². The parent function in f(x) = ln (x - 6) is ln

Perfect Square Trinomial (minus version) (a - b)²

Think of it like (a-b)(a-b)so... (a - b)² = a² - 2ab + b²

What is inverse of f(x) = 5x -1

To find inverse, you undo the basic function. Do the operations in reverse order from normal. So, first undo the subtraction and then undo the multiplication. So, the inverse is (x +1)/5.

correlation coefficient

a number between −1 and +1 calc lated so as to represent the linear dependence of two variables or sets of data.

What is the Standard Form for Quadratic function (parabola)

f(x) = ax² +bx + c

how to maximize and minimize

find the vertex points on a graph (where any of the lines intersect then plug the points into a given equation and the biggest is the maximum and the smallest is the minimum

constraint

linear inequality

what is ln a

ln stands for natural log. So ln a is just the log, with base e, of a.

log₃3⁶⁺ⁿ = ?

log₃3⁶⁺ⁿ = 6 + n (When base and a are same, and there is an exponent of a, the answer is just the exponent). This is because you can pull the exponent out and make it a coefficient, and are left with log₃3, and log in base x of x is always 1, so you get the coefficient (i.e. the original exponent (6 + 1)) times 1

log₃81 = ?

log₃81 = log₃9² = log₃(3²)² = log₃3⁴ = 4

Describe transformation and asymptote for equation p(x) = 2 logx - 1

p(x) = 2 logx -1 Parent function is f(x) = log x Asymptote is x = 1 Transformation is vertical stretch by factor of 2 (because of coefficient of 2)., The subtracting one causes a vertical shift down by 1 unit,

Why does log function used for pH measurements

pH is a measurement of the number of hydrogen atoms in a given volume of liquid. the range of hydrogen atoms is huge, and may go from 100 to 100 billion. to come up with a handy scale to measure pH we don't want to have to use numbers with that big a range, so we use a logarithmic scale, where we are looking at the exponents needed to generate those numbers, rather than those numbers themselves. So, pH ranges from 1 to 14, but it's a common log (i.e. base 10) function. So, it's really saying that we're multiply 10 times itself from 1 to 14 times to get the scale.

feasible region

shaded region on a graph

absolute value

the value of a real number without regard to its sign.

x-intercept

the x-intercept is the x value when y is equal to 0 or is the x value where our graph actually intercepts the x axis.

axis of symmetry for quadratic function in vertex form

x = h (Because when f(x) = a(x - h)² + K, the way to make the thing that you're squaring to be zero is when x = h)

How do you find the inverse of a function

you "undo" the operation of the original function. For example, start with a basic function of f(x) = x/3 (i.e. the basic function is that y = x divided by 3). The inverse of that function is to "undo" that operation, so instead of dividing by 3 you would multiply by 3. ƒ⁻¹ is how we write the inverse of function f. So, if f(x) = x/3, then f⁻¹(x) = 3x

How do we know if something is exponential growth or exponential decay. Growth first.

Remember basic exponential formula is f(n) = abⁿ. If b > 1, then the function is exponential growth, because when you multiply a number greater than 1 by itself more and more times, the resulting number gets bigger and bigger. 2² = 4. 2³= 16. 2⁴ = 32.

What is the y intercept

Where the function crosses the y axis. In other words, when x is zero, what is the y coordinate.

What is quadratic function

Where the x factor has a square function. Example f(x) = x². Creates a parabola shaped graph.

Turning exponent into coefficient - does it work both ways

Yes, it works both ways. You can move the coefficient into the exponent, or move the exponent into the coefficient. 3log aⁿ = log a³ⁿ log aⁿ = n log a

What is formula to find the rate of constant growth or decay?

f(n) = a(1± r)ⁿ. Where r is the rate of change, and n is the number of periods, and a is the starting point. Example - start with 100 students in a class, grow that number by 3% each year, and let that happen for 5 years. f(n) = 100(1+.03)⁵ = 100 (1.19) = 119

What is basic form of exponential function

f(n) = abⁿ The variable (here, n) is the exponent. So, as we move along the x axis, we're changing the number of times we're multiply b times itself, which creates a very steep curve.

linear programing

finding min/max of a function

What is logarithm function part 2

if bⁿ = a, then n is the exponent that we have to raise b to in order to get a. So, basically making n the output instead of the input. It's fixing the base and the outcome, and asking what exponent do we need to make the equation work. so, 2³ = 8. So, log₂8 = 3. (Note - in their examples, they use x instead of n but this program won't put an x in the exponent).

What is formula to change the base of a logarithm?

log₃x = log₂x/log₂3 So, you can change the base, as long as you then divide by the log (with the new base) of the old base

Use quotient property to solve this: log₅625 - log₅125 = ?

log₅625 - log₅125 = log₅(625/125) = log₅5 = 1

Use product property to solve log₆4 + log₆9 = ?

log₆4 + log₆9 = log₆(4 x 9) = log₆36 = 2

linear system

to solve the it is where the lines intersect can find my elimination and substation (shown above)

What is the Y intercept of Quadratic Function in Standard Form

when f(x) = ax² + bx + c, the Y intercept is found by making x = 0, so it is y = c. So, the coordinates are x = x; y = c. We write this as (0,c)

elimination system

by subtracting one system in order to eliminate one variable {2y+x=8 {-y-x+-3 just add these two because it will result in one variable ∴ {y=5 then you would plug that back in to the previous equations to get x=-2

What is the Vertex Form for quadratic function

f(x) = a(x - h)² + k

What is the Axis of Symetry for quadratic equation in Standard Form (f(x) = ax² + bx + c)

Axis of symmetry is x = -b/2a

When a Quadratic Function is in Vertex Form, what are the coordinates of the vertex?

(h,k)

What is the inverse of a function

A basic function takes an input ("x") and generates an output ("y"). For every x value, there is a corresponding y value. We say "Y is a function of X." In an inverse, we switch the relationship. So, what was the input (x) becomes the output (y), and what was the output (y) becomes the input (x) It looks like a mirror image of the graph, with the line of symmetry at the x = y line.

What is exponential function

A function where the base is a constant -- and the exponent varies -- is called an exponential function. The standard form is f(x) = abⁿ (but x instead of n as the exponent). a is the co-efficient. b is the base. x (or n here) is the exponent.

consistent system

A system of two linear equations can have one solution, an infinite number of solutions, or no solution. for an inconsistent solution it would have one solution. where the lines intersect at exactly one point.

In an exponential decay formula, what is inside the parenthesis?

Basic formula for constant rate of change is f(n) = a(1± r)ⁿ. Where r is the rate of change, and n is the number of periods, and a is the starting point. When it is decay rather than growth, the r is negative. So, if the change is that something is losing 3% each year, r would be -.03. So, inside the parenthesis would be (1 - .03) = .97. Since this is less than 1, we have decay rather than growth.

Take a parent function of f(n) =0.5ⁿ, and translate it 3 units to the left, compress vertically by a factor of ¼ and reflect across the x axis

Begin with parent function: f(n) = (0.5)ⁿ To translate 3 units to the left, replace n with n + 3. To compress vertically by 1/4, multiply the function by 1/4 To reflect across the x axis, make the function negative So, g(n) = - ¼(0.5)ⁿ⁺³

Finding y intercept

In general, since the y intercept is where x is zero, you find it by making x = 0 and solving equation. (in other words, just plug in 0 for x and solve.)

What is horizontal stretch or compression

Changing the shape of the curve in the horizontal dimension is done by changing the x value before the function is performed. f(x) = f(1/bx) Example: y = 2¹/⁵ⁿ is stretch, because the exponent is less than one. y = 2³ⁿ is compression because the exponent is greater than 1. The horizontal translation is a stretch if the 1/b factor is less than 1. The horizontal translation is compression if the 1/b factor is greater than 1.

What is the Difference of two squares, and how do you simplify it when you see it

Difference of two squares means you are subtracting something squared from something else squared. So, it's like a² - b². You can always simplify it into the following: a² - b² = (a + b) (a - b)

How can you use completing the square to put equations into vertex form

Remember, vertex form is f(x) = (x - h)² + k (where the vertex is point (h, k)) So, you can use completing the square to give you a perfect square binomial (x - h)² Example -- f(x) = x² + 10x - 13. Here are the steps: 1. Set up to complete the square: f(x) = (x² + 10x + __) - 13 - __ (Just moving terms around and creating blank space where we will add (and subtract) (1/2 * b)² (aka (b/2)² b is 10 (from the 10x element in the original formula) so (b/2)² is (10/2)² = 100/4 = 25 2. f(x) = (x² + 10x + 25) - 13 - 25 Factoring the perfect square we created inside the parenthesis gives us: f(x) = (x + 5)² - 13 -25 So, f(x) = (x + 5)² - 38 Now it's in In vertex form, so h = -5 and k = -38, so the vertex is (-5, -38)

y-intercept

The equation of any straight line, called a linear equation, can be written as: y = mx + b, where m is the slope of the line and b is the y-intercept. The y-intercept of this line is the value of y at the point where the line crosses the y axis. That's where x = 0. So, when in y = mx + b form, the y intercept is b, since that's what's left of the equation when x = 0.

How do you find the Y intercept for quadratic equation in vertex form

To find the Y intercept, just make x = 0 and solve the equation. so take f(x) = a(x - h)² + k, and put 0 in for x and solve.

solve log 30x - log 6 = 4

Use quotient property first. If subtracting logs, you can take the log of the quotient instead. So,log 30x - log 6 = 4 log (30x/6) = 4 log (5x) = 4 undo the logs, using base 10 10 to the 4th power = 5x 10000 = 5x x = 2,000

What does it mean for an equation to be linear? What shape does the graph look like and why?

Linear means none of the variable are squared (or higher power). Just some x's and y's, but no x² or y². The graph is a straight line (that's why it's called "linear"). It's that shape because x varies with y in a linear fashion. Maybe have different slope or y intercept, but always a straight line because it's always just y = mx + b.

What is log₃3?

Log base anything of that same thing is always 1. Log is just the exponent. So, the exponent you need to use to raise 3 by something to get 3. It's always one

What is logarithm function

Logarithm is the inverse function of an exponential function. For example, take a basic exponential function of 3ⁿ = a. The logarithmic equation of that is log₃a = n

How to solve equation where the unknown is in the exponent. Example - 27ⁿ = 3ⁿ⁺⁸

First, try to put everything in same base. Here, 3 will work because both 27 and 3 can be expressed with a base three and an exponent of something. 27 = (3³) so 27ⁿ = (3³)ⁿ So, (3³)ⁿ = 3ⁿ⁺⁸. Once you have the same base (3) -- then exponents have to be equal. Example -- xⁿ⁺³ = x²ⁿ then n+ 3 has to equal 2n. so n+3=2n, so you can solve it. Back to the top example. Once we see that (3³)ⁿ = 3ⁿ⁺⁸, we know that 3n = n+8 so 2n = 8 so n = 4.

Solve log x + log (x+3) = 1

First, use product property log x + log (x+3) = 1 log x(x+3) = 1 Use 10 as base and "exponentiate" (i.e. make the expressions the exponents with 10 as base 10 to the power of log x(x+3) = 10¹ put it all on one side and create quadratic equation x² + 3x -10 = 0 (x-2)(x+5) = 0 So, it looks like x could be 2 or -5 But with logarithms, you need to check possible answers to see if they can in fact work (or else they are called extraneous answers if they can't). Plug the answers back into original equation. If x = 2, log2 + log 5 = log 10, which is 1 so that's fine. If x = -5, we'd be taking log of a negative number which we can't do. There is no power to raise 10 to to get a negative number. So, can't have negative log. So, x=-5 is extraneous solution and only real answer is x=2

What is quotient property of logarithms

For exponents, when you divide two polynomials, you subtract the exponents. b⁸/b³ = b⁸⁻³ = b⁵. For logarithms, same thing. log₃(m/n) = log₃m - log₃n

What are the First Steps for completing the square

Get all the x and x² factors on one side, and everything else on the other side. Get the x² factor to 1 (so for example if it's 6x², divide everything by 6). So it looks like this: x² + bx = c then, add (b/2)² to both sides (i.e. take half of b, then square it, and add that to both sides. This will give you a perfect square on left side, so you can later square root both sides. Go to step 2 on separate card

When you raise a number to a power which is itself a log, with the same base as the first number, the answer is just the output number in the exponent. (This is the tongue twister principle)

If the exponent is log₄9 and we're raising 4 to that power, the answer is simply 9. The whole meaning of the exponent log₄9 is "what power (exponent) do I have to raise 4 to in order to get 9." So, if I am in fact raising 4 to that power, I get 9. Simple way is to say the 3 to the power of log₃9 is that 3 and the log₃ exponent cancel out, leaving just 9.

What is a vertical transformation?

If you add or subtract a constant to the parent function, you just move the graph up or down the y axis by that amount. (It's like the b in the y = mx + b linear formula; it doesn't impact slope; it just moves the line up or down and impact the y intercept So, if function is y = 2ⁿ + 3, you have a shape defined by the parent function (y = 2ⁿ) (Where n is x), with a vertical translation of being moved up 3.

y co-ordinate for vertex for quadratic function in vertex form

K (because in the equation f(x) = a(x - h)² + K, the vertex is on axis of symmetry, meaning x=h and whole first piece of equation is zero, so what is left of equation is f(x) = K. So, that's the y coordinate of the vertex.

What does it mean for an equation to be quadratic? What shape does the graph look like and why

Quadratic means there is an x² component. It looks like a parabola, which means that it has axis of symmetry, so there a point (the vertex), and things on the left side of the vertex are the mirror image of things on the right side of the vertex. This is because we are squaring x, and when you square a negative number you get the same result as when you square the same number but positive. Example: if x is 2, x² = 4. If x is -2, x² = 4. So, as you move two units away from the vertex, in either the negative or positive direction, you get the same y value (4). That's why it's symmetric.

What is a conjugate pair

Relation of two complex numbers a ± bi is a conjugate pair of a + bi and a−bi. In a conjugate pair, the real number is the same and the imaginary parts are opposites.

How do we know if somethings exponential decay?

Remember basic exponential formula is f(n) = abⁿ. If b < 1, then the function is exponential decay, because when you multiply a number less than 1 by itself more and more times, the resulting number gets smaller and smaller. ½² = ¼. ½³= 1/16. ½⁴ = 1/32.

How do you solve log equations

Remember the log is the exponent. It is the exponent that we have to raise the base to in order to get the output. You write it with the base in small letter/number below the word "log." the output is the next number, and the exponent is on the other side of the equal sign. So, use 3 as example of base. log₃a = x means you have to raise 3 to the x power to get a. So, log₆36 = 2, because you raise 6 to the second power (squared) to get 36.

Describe transformation and asymptote for equation q(x) = -ln(x+2)

Start with function: q(x) = -ln(x+2) Parent function is q(x) = ln x Asymptote is x = -2, because that is what x has to be in order to be taking the natural log of 0 (Remember, asymptote is when you are taking the log of 0, because it never quite gets there - as you use smaller and smaller exponents, you can get closer and closer to 0, but never quite reach zero. So, the asymptote is where the expression (x+2) is zero, i.e. where x = -2. So, transformation is vertical shift 2 units to the left. Also, since it is negative ln, there is reflection across the x axis. Remember basic formula log₃a = n means 3ⁿ = a. a can never quite get to zero, because there is no exponent (n) that will reduce a positive base to zero. It can get very small, but not all the way to zero. (Remember, I just use 3 for the base (b) because the software won't let me put anything except numbers down below)

Use change of base theory to solve log₂₇81

Use base three, since both 27 and 81 can be obtained by exponenting 3. log₂₇81 = log₃81/log₃27 = 4/3 = 1.33333

Basically, what is the log

The log is the exponent that we need to raise a given base to (b), in order to yield an outcome (a). Example: 2³ = 8. So, log₂8 = 3. That's telling us that we need to raise the base (2) to the third power in order to achieve the result of 8. The base is written smaller and below the word "log" (here, 2) and the outcome is written normally after log (here 8). Then you solve for the exponent (here, 3)

What are the roots of a quadratic equation?

The roots are when you set the quadratic expression equal to zero, and then solve for x. You usually do this by factoring the expression to get it into a form such as (x +a)(x + b). then the roots would be -a and -b, since if x is either of those values, the whole expression would equal 0. Example - to find the roots of 9x² - 1 = 0. First, factor 9x² - 1 into (3x + 1)(3x - 1). So, (3x + 1)(3x - 1) = 0. that means that either 3x + 1 = 0 or 3x - 1 = 0. (Since multiplying anything times 0 gives you 0. So, 3x + 1 = 0 → x = -1/3 or 3x - 1 = 0 → x = 1/3 So, the roots are -1/3 and 1/3 aka ±1/3

What does a table of a linear function look like

The values of y go up at a constant rate. For example, as x goes up 1 if y always goes up 3, that means you have a linear function with a slope of 3. y = 3x + B.

What is reflection over the x axis?

We can reflect over the x axis with y = -f(x) This keeps the function the same, but just turns the results into their opposite (positive values become negative values, so what was a graph shooting up in now a graph shooting down). This reflects it over the x axis because it is just reversing the y values from the original function. Example : y = -2ⁿ

What is reflection over the y axis

We can reflect over the y axis with y = f(-x) This reverses the function, because it is changing the input, not the output. So, instead of a positive 1 x value giving us a certain y value, the negative 1 x value is giving us that y. So, the graph is being flipped around the y axis because it is switching the inputs instead of the outputs. Example: y = 2⁻ⁿ

What is a "common logarithm"

When the base is 10, you don't write it. So, log 100 = x means the base is 10, the result is 100, and x is the exponent that will turn 10 into 100. So, since 10² = 100, the log is 2. So, log 100 = 2.

What is the natural logarithm

When the base is the special constant e, that is called the natural logarithm. e = 2.71828.... ln means natural log. So ln a means the natural log of a (with base is e)

Solve log₄(x-1) = 3

When the unknown (x) is in the a part of the basic log a = x formula, just use the definition of log functions, because we know the base and we know the exponent (x). So, just raise the base to the exponent, and put that on one side of the equal sign, and put the A expression (including the unknown) on the other side. So, 4³ = (x-1) So 64 = x-1. So, x = 65.

What is the square root property

You can take the square root of both sides of an equation to help solve a quadratic equation. Be sure to include both the positive and negative square roots. x² = 15 x = ±√15 Note: this is why we sometimes use completing the square to solve quadratic functions -- because if we can get one side of the equation to be a perfect square (for example, (x + 4)² then we can square root both sides of the equestion.

Second steps for completing the square

after adding (b/2)² (in other words, one-half b, squared) to both sides -- Factor the left side into a perfect square (i.e. something squared), then take square root of both sides, and do the clean up. Example: x² + 6x = 27 (given, already moved to correct format) x² + 6x + 9 = 27 + 9 (adding (1/2 of 6)² to both sides) (x+3)² = 36 (factoring left side into perfect square) x+3 = ± 6 (square root both sides) x = 3 ± 6 = 9 or -3 plug back in to double check

Finding zeros with perfect square approach sometimes leads to complex numbers

complex number can be written in form a + bi Example: f(x) = x² - 2x + 5. To find the zeros with perfect square strategy, here are steps: 1. set equal to zero. x² - 2x + 5 = 0 2. Rewrite for perfect square strategy → x² - 2x + __ = -5 + __ 3. Fill in both blanks with (b/2)² (here, (2/2)² = 1 So, x² - 2x + 1 = -5 + 1 x² - 2x + 1 = -4 4. Factor (x - 1)² = -4 5. Sqaure root both sides x - 1 = ±√-4 6. Simplify. x = 1 ±2i

What is standard formula for vertical transformation

f(x) + k, when f(x) is the parent function.

objective function

function you want to maximise

Take a parent function of f(x) = ln x and translate 5 units to the right and 2 units down, and horizontally stretch by a factor of 3

g(x) = ln (x/3 -5) - 2

What is horizontal transformation

horizontal translation is where the x values are changed. So instead of just f(x), you change the x value (the input) and create movement along the x axis (horizontal movement). So, it's f(x-h). For example, y = 2ⁿ is the parent function, but y = 2ⁿ⁻² would be the same shape, just moved 2 units to the right. (Negative numbers move it to the right; positive numbers move it to the left)

What is the power property of logarithms?

log₃aⁿ = n log₃a The exponent (n) pulls out and becomes a coefficient exponent analog - when you take a number that already has an exponent, and exponent it again, you multiply the two exponents. So, a² to the third power is a² x ³ = a⁶

solve log3x² = 8

log₃x² = 8 move exponent to coefficient: 2log₃x = 8 divide by 2, so log₃x = 4 From here, two ways to solve: 1. Use inverse function, by making these expressions the exponents with a base of 3. So, 3 to the power of log₃x = 3⁴ Taking three to the power of log₃x, the 3 and log₃ piece cancel out, leaving you with x on that side. x = 3⁴ = 81 or, method 2 (this is easier way) -- use definition of logarithm (chant : logarithm is the exponent) log₃x = 4 means that if 3 is the base and 4 is the exponent, then x is the output. So 3⁴ = x and x therefore = 81

What is log₄64⁵ ?

log₄64⁵ = 5log₄64. We know that log₄64 is 3, because we raise 4 to the 3rd power to get 64. So, answer is 5(3) = 15

log₆(1/36)⁴ = ?

log₆(1/36)⁴ = 4log₆(1/36) = 4(-2) = -8

system of equations

set of two or more variables or equations {x+2y=10 {3x-y=9 is it (4,3) yes when you plug in the (x,y) it would be correct

What is the quadratic formula?

the Quadratic formula is a way to find the zeros of a quadratic function, where that function can't be easily factored. Example: Find the zeros of f(x) = x² - x + 2 Steps: 1. Set equal to zero. x² - x + 2 = 0 2. Write quadratic formula. x = (-b ± √b² - 4ac)/2a 3. Fill in the a, b and c's x = (-(-1) ± √(-1)² - 4(2)(2)) / 2(2) x = (1 ± √1 - 15) / 4 x = (1 ± √-15) / 4 x = (1 ± i √15) / 4 x = 1/4 ± (√15/4)i

vertex printable of linear programing

the max/min that occurs at the vertices