Pretest: Exponential Expressions and Equations
The population of a small town is decreasing exponentially at a rate of 14.3% each year. The current population is 9,400 people. The town's tax status will change once the population is below 6,000 people. Create an inequality that can be used to determine after how many years, t, the town's tax status will change, and use it to answer the question below. Will the town's tax status change within the next 3 years?
1. 9400 2. .8957 3. 6000 4. yes Explanation: The amount of increase is 100 - the amount of decrease. The amount of increase = 1 - .143=.857 y=9400*(.857)^t (t is an exponent) You can graph this by replacing t with an "x." Then look at the point (x, 6000), what is the value of x? OR Plug in the value 3 for t and calculate this: 9400 * (.857)^3 (three is an exponent) = 5916. This is truly below 6000, so the answer is, yes, the tax status will change.
It is a widely held belief that a typical piece of printer paper with a thickness of 0.1 mm can go through at most 7 folds. When folding a piece of paper in half, the number of layers doubles with each fold, which also doubles the thickness. Sylvia is testing this theory and employing a hydraulic press in order to ensure each fold is as tight as possible. If Sylvia is successful in folding the paper completely, determine the thickness of her piece of printer paper after being folded in half 7 times.
Answer = 12.8 The equation being used here is: y = .1 (2) ^ x-1 Where y = the thickness in mm and x represents the number of folds. You can graph this and look at the value of y at x=7. Using a calculated, create a table of values: x = 1, y=.1 x=2, y=.2 x=3, y=.4 x=4, y=.8 x=5, y=1.6 x=6, y = 3.2 x=7, y=6.4 x=8, y=12.8 Or, easiest, multiple .1 times 2, 7 times (for the 7 folds): .1 x 2 x2 x2 x2 x2 x2 x2 = 12.8mm
Which exponential equation is equivalent to this logarithmic equation? log5 x−log5 25 =7
Answer: 5^9 = x A. Explanation: Using the rules for logarithms. When you subtract, those combine to be a quotient. Log base 5 of x minus log base 5 of 25 simplifies to: Log base 5 of (x/25) A log always equals an exponent, and the base, well that's the base, so: 5 to the 7 power equals x/25. 25 = 5^2, if you move this to the numerator on the right side of your equation, the exponent is now negative: 5^7 = 5^-2 (x) Divide both sides by 5^-2. 5^7 / 5^-2 simplifies to 5 ^(7--2) = 5^9. Remember, quotient rules for exponents - top minus bottom. This turns the 2 positive when you simplify. Minus a negative = Plus. So, 5 to the 9th power = x.
Which expression is equivalent to the given expression? 4lnx+ln3−lnx
Answer: C. ln(3x^3) Explanation: 4(ln(x)) = ln(x^4) is the first piece. This is add to ln (3) which turns it into a multiplication: ln (x^4) + ln (3) = ln(3* x^4) This is part of a subtraction now, which turns this into a division problem, according to the rules for natural logs; ln(3x^4) - ln(x) = ln(3x^4/x) 3x^4/x simplifies to 3x^3
What is the solution set to this equation? log2(x−2)+log2x=3
Answer: C. x=4 Explanation: Use the logarithm rules to combine the logs. Remember addition means to combine with multiplication. logbase2[(x-2)*(x)] = 3 logbase2(x^2-2x) = 3 2^3 = x^2 - 2x 8 = x^2 -2x Subtract 8 from both sides. 0 = x^2 -2x - 8 0 = (x-4)(x+2) Solutions 4 and -2 BUT, you have to verify the solutions in the equation. -2 is not allowed. You can't take the log of a negative number. The only solution is 4.
Which logarithmic equation is equivalent to this exponential equation? 6x=216
Answer: D. log6 (216) = x Explanation: exponent = x base = 6 Remember how you read a log: The log base (?) of (?) = an exponent. log base 6 of 216 = x
Which exponential equation correctly rewrites this logarithmic equation? log6 18 = x (log base 6 of 18)
Answer: 6^x = 18 (C)
Jackson needs to determine the value of x in this equation. Rewrite the expression as a logarithmic quotient that he could enter in his calculator. 1.13x=2.97
Answer: log base(1.13) of 2.97 = x Explanation: 1.13 to the x power = 2.97. The base is 1.13. A log always = the exponent. The exponent here is x. So, the log (subscript-lower-the base)1.13 of 2.97 = x
Power usage is measured in kilowatt-hours, kWh. After 7 a.m., the power usage on a college campus increases at a rate of 21% per hour. Prior to 7 a.m., 15,040 kWh have been used. The university has a daily goal to keep their power usage less than or equal to 100,000 kWh. Which inequality can be used to determine the number of hours, t, after 7 a.m. when the power usage on campus will be less than or equal to 100,000?
Answer: 15,040(1.21)^t <= 100,000 Explanation: The initial value = before 7am usage = 15,040 The percentage of increase = 1+.21 = 1.21 The number of hours = t (or x in the standard exponential form equation)
David invested $230 in a savings account that offers a 3% return on the investment. The value of David's investment will be at least $415 after a period of years. Hint: Use the formula A = P(1 + r)^t, where A is the amount after t years, P is the amount invested, r is the rate of interest, and t is the time period. Use a calculator to compute the answer, and round it off to the nearest year.
Answer: 20 Explanation: A is the amount we are wanting = 415 P is the amount we start with = 230 The interest rate is 3% = .03 Equation: 415 < 230(1.03)^t You could plug in values for the year of 1, 2, 3, etc. until you reach 415 or above. OR solve the equation, using an equal sign instead of the inequality: 415 = 230(1.03)^x Divide both sides by 230. 1.80 = 1.03^x Take the ln of both sides ln(1.8) = x * ln(1.03) Divide both sides by ln(1.030 ln(1.8)/ln(1.03) = x x = 19.8855, rounded to nearest year = 20
Tests show that the hydrogen ion concentration of a sample of apple juice is 0.0003 and that of ammonia is 1.3×10-9. Find the approximate pH of each liquid using the formula pH=-log[H+], where [H+] is the hydrogen ion concentration. The pH value of the apple juice is ___________. The pH value of ammonia is _____________.
Answer: 3.5, 8.9 Explanation: Use the rule -log(a) = log(1/a) a=the concentration of hydrogen or [H+] For apple juice, this is given as .0003 Take the log of 1/.0003 in your calculator. = 3.522 Ammonia: H+ = 1.3 x 10^-9m since it's a negative exponent, move your decimal LEFT 9 times = .0000000013 Take the log of 1/.0000000013 = 8.9 Then ask yourself it this makes sense. The pH of apple juice is lower than ammonia. This means that apple juice is acidic (lower than 7) and ammonia is basic (higher than pH). Yes, it makes sense. :)
What is the solution to this equation? 3^2x+12=39
Answer: 3/2 Explanation: 3^2x+12=39 3^2x+12−12=39−12 3^2x=27 2x* ln3 = ln27 Divide both sides by ln 3 and use your calculator. 2x = ln27/ln3 2x=3 x = 3/2
What value of x satisfies this equation? log(2x)=2
Answer: 50 Explanation: The base here is an understood ten. It's a plain old base 10 log. Rewrite this: 10^2 = 2x 100 = 2x x=50 IF x was 50, then, double check your equation: log(2*50) = log 100, put this in your calculator to double check.
What is the value of x? log3x=4
Answer: 81 Explanation: 3 to the 4th power = what this in your calculator: 3^4
What is the solution to this equation? 216=6^2x−1
Answer: A. x=2 Explanation: Take the ln of both sides. ln216 = (2x-1) * ln 6 Divide both sides by ln6. Put the quotient ln216/ln6 in your calculator. 3 = 2x-1 Add one to both sides and divide both sides by 2: x = 2
What is the solution to this equation? ln(2x+4)=ln(x+3)
Answer: C. -1 Explanation: Since the ln is being taking on both sides, and they're equal, the items in parenthesis are equal. 2x + 4 = x + 3 Solve for x by subtracting 4 from both sides and subtracting x from both sides: x = -1
A population of pea aphids doubles every week. The population begins with 300 individual pea aphids. In how many weeks will the population reach 9,600 individuals?
Answer= 5 weeks Explanation: You can always multiple 300 x 2, and count the number of times you have to do it to get to 9600, that's the number of weeks. (It takes 5 times=5weeks) The equation is: y = 300(2)^x (x is an exponent) Graph this equation and find where y = 9600, the x coordinate for that point is your answer. Quickest and easiest, easier than solving this exponential equation, is to plug in the 4 values of x (the number of weeks) into the equation and see which one gives you 9600.
Rewrite the given equation in logarithmic form. Then, select all of the equations with an equivalent solution. 8e^x − 5 = 0
Answers: x = ln(5/8) AND x = ln5 - ln 8 Explanation: Add 5 to both sides: 8e^x = 5 Divide both sides by 8. e^x = 5/8 Simplify, using ln since we are dealing with e. ln(5/8)=x x = ln5 - ln 8
What is the solution to this equation? (1/4) ^x+1 = 32
C. -7/2 Explanation: The trick here is to add one to your answer choices. They've given you the value of x as choices. Plugging in -7/2 into the equation, we get: (1/4) ^-7/2 + 1 (the up arrow means that's the exponent) (1/4) ^ -5/2 = 32
The area of undeveloped land in a suburban town is decreasing at a rate of 17.3% annually. In 2016, there were 3,400 acres of undeveloped land. If t represents the number of years since 2016, which equation can be used to determine after how many years the town will have 900 acres of undeveloped land remaining?
Equation: y = 3400 ( 1-.173) ^x (x is the exponent) The value of y they are asking about is the amount of land left = 900 ANSWER: 900 = 3400(.827)^t (t is the exponent)