Maths - Exponentials and Logarithms

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What may the natural logarithm also be called?

The Naperian logarithm

What can the number in the logarithm (not the base) be called?

The argument

The larger the base of the exponential function...

The closer the curve comes to the y-axis

What does this mean in terms of graph transformations?

The graph of y = a^x reflected on the line of y=x = log(base a)x

What is the relationship between a^x and a^-x

The graphs are flipped in the y axis

What is meant by index form?

The opposite of logarithmic form, using exponents

What is a general rule for using bases in logarithms?

They cannot be negative

Since this is true, what can be said about the intersections of a logarithmic function its respective exponential function?

They will intersect on the line y=x

If there's no mathematical reason as to why a model may be unsuitable, what should be done?

Use a qualitative argument

Prove this formula

Suppose x = a^y log(base n)x = y log(base n)a y = log (base n)x/log(base n)a log(base a)x = log(base n)x/log(base n)a

How should you begin by solving some logarithms?

Take logs on both sides

What is the general equation for an exponential function?

y = a^x where a is a positive constant

How can this be shown?

y = a^x x = a^y log(base a)x = y y = log(base a) x

Give two popular relationships and their names

y = axⁿ y = ke^x

The most common base is...

10

What is the approximate value of e and what is the nature of the number

2.718 Irrational

What is the inverse of an exponential function?

A logarithmic function

Therefore, what is the derivative of the function e^kx

ke^kx

What relationship can therefore be stated?

ke^kx is directly proportional to y, where k is the constant of proportionality

Give four key points of logarithms

log (base 10) can be expressed as log a¹ = a therefore log (base a) (a) = 1 a⁰ = 1 therefore log (base a)(1)=0 a⁻¹ = 1/a therefore log (base a)(1/a) = -1

What is the second law of logarithms?

log(base a)(x/y) = log(base a)(x) - log(base a)(y)

What is the first law of logarithms?

log(base a)(xy) = log(base a)(x) + log(base a)(y)

What is the third law of logarithms?

log(base a)(xⁿ) = nlog(base a)(x)

Give the change of base formula for logarithms

log(base a)x = log(base n)x/log(base n)a

Construct a proof for the third law of logarithms

log(base a)xⁿ = log(base a)x1*x2*x3*x4...xn multiplication rule states that: log(base a)xⁿ = log(base a)x1 + log(base a)x2... Therefore = nlog(base a)x

What point do all exponential functions pass through and why?

(0,1) No matter the base, any number to the power of zero is one

What point do all logarithmic functions pass through and why?

(1,0) In any base, to get any number to equal one, you must raise it to 0

How else can a^-x be expressed

(1/a)^x

Why are exponentials often used as mathematical models?

Because the rate of change of some variable is proportional to the value of the same variable

When asked if data fits a mathematical model, what must be done?

Calculate percentage difference in data and mathematical value, base assumptions off of that

A base raised to an exponent and its logarithmic form can be said to be...

Equivalent statements

What are asymptotes of exponential and logarithmic functions?

Exponential: y≠0 (undefined) Logarithmic: x≠0 (undefined)

How do you solve logs when there are multiple operations?

Go from left to right

What is one reason a model may not appropriate?

If the variable is discrete

What happens to the shape of an exponential function as you increase the base?

It bends to the left

What happens to the shape of a logarithmic function as you increase the base?

It curves to the right

What is the name of ∞?

Lemniscate

How is this read?

O is the log of x to the base n

How would you model a compound interest after t years using an exponential function?

P(100+A/100)^t Where A is percentage and P is initial price

What can you do when you have logs on both the numerator and denominator?

Simplify them to common logs and cancel out

How do you prove that a model shows an upper limit

Simplify to get a whole number on the top, go from there

Give the standard logarithmic form of any base raised to an exponent

Where n°=x logₙ(x) = O

When a logarithmic problem is deduced to a quadratic/cubic, what must be kept in mind?

You cannot have a negative solution of x because you cannot take the negative of a log

Why can you not take the logarithm of a negative number?

You cannot raise a positive integer to something to get a negative number

These only work for what cases?

a > 0, a ≠ 1

Construct a proof for the second law of logarithms

a^n = x a^m = y log(base a)(x) = n log(base a)(y) = m x/y = a^n / a^m = a^(n-m) log(base a)(x/y) = n-m log(base a)(x/y) = log(base a)(x) - log(base a)y

Construct a proof for the first law of logarithms

a^n = x a^m = y log(base a)(x) = n log(base a)(y) = m xy = a^n * a^m = a^(m+n) log(base a)(xy) = m+n log(base a)(xy) = log(base a)(x) + log(base a)y

How can e be represented as a limit?

e = lim n-> ∞ (1+1/n)^n

Give a rule about the transformation of e^x

e^kx will compress the function by a scale factor of k However, for the same value of y, the transformed function will have a gradient of ke^x

What is the exponential function and why?

e^x The derivative of this graph is equal to itself

How else may e^x be written?

exp(x)


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