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Chapter 6 Exponential and Logarithmic Functions

Chapter 6 Practice Test

Practice Test

  1. The population of a pod of bottlenose dolphins is modeled by the functionA(t)=8(1.17)t, wheretis given in years. To the nearest whole number, what will the pod population be after3years?
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About13dolphins.

  1. Find an exponential equation that passes through the points(0, 4)and(2, 9).
  1. Drew wants to save $2,500 to go to the next World Cup. To the nearest dollar, how much will he need to invest in an account now with6.25APR, compounding daily, in order to reach his goal in4years?
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$1,947

  1. An investment account was opened with an initial deposit of $9,600 and earns7.4interest, compounded continuously. How much will the account be worth after15years?
  1. Graph the functionf(x)=5(0.5)xand its reflection across the y-axis on the same axes, and give the y-intercept.
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y-intercept:(0, 5)

Graph of f(x)=5(0.5)^−x and f(x)=5(0.5)^x

  1. The graph shows transformations of the graph off(x)=(12)x.What is the equation for the transformation?

The graph shows transformations of the graph of f(x)=(1/2)^x

  1. Rewritelog8.5(614.125)=aas an equivalent exponential equation.
Show Solution

8.5a=614.125

  1. Rewritee12=mas an equivalent logarithmic equation.
  1. Solve forxby converting the logarithmic equationlog17(x)=2to exponential form.
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x=(17)2=149

  1. Evaluatelog(10,000,000)without using a calculator.
  1. Evaluateln(0.716)using a calculator. Round to the nearest thousandth.
Show Solution

ln(0.716)0.334

  1. Graph the functiong(x)=log(126x)+3.
  1. State the domain, vertical asymptote, and end behavior of the functionf(x)=log5(3913x)+7.
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Domain:x3;Vertical asymptote:x=3;End behavior:x3,f(x)andx,f(x)

  1. Rewritelog(17a2b)as a sum.
  1. Rewritelogt(96)logt(8)in compact form.
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logt(12)

  1. Rewritelog8(a1b)as a product.
  1. Use properties of logarithm to expandln(y3z2x43).
Show Solution

3ln(y)+2ln(z)+ln(x4)3

  1. Condense the expression4ln(c)+ln(d)+ln(a)3+ln(b+3)3to a single logarithm.
  1. Rewrite163x5=1000as a logarithm. Then apply the change of base formula to solve forxusing the natural log. Round to the nearest thousandth.
Show Solution

x=ln(1000)ln(16)+532.497

  1. Solve(181)x1243=(19)3x1by rewriting each side with a common base.
  1. Use logarithms to find the exact solution for9e10a85=41. If there is no solution, write no solution.
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a=ln(4)+810

  1. Find the exact solution for10e4x+2+5=56.If there is no solution, write no solution.
  1. Find the exact solution for5e4x14=64.If there is no solution, write no solution.
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no solution

  1. Find the exact solution for2x3=62x1.If there is no solution, write no solution.
  1. Find the exact solution fore2xex72=0.If there is no solution, write no solution.
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x=ln(9)

  1. Use the definition of a logarithm to find the exact solution for4log(2n)7=11
  1. Use the one-to-one property of logarithms to find an exact solution forlog(4x210)+log(3)=log(51)If there is no solution, write no solution.
Show Solution

x=±332

  1. The formula for measuring sound intensity in decibelsDis defined by the equationD=10log(II0),whereIis the intensity of the sound in watts per square meter andI0=1012is the lowest level of sound that the average person can hear. How many decibels are emitted from a rock concert with a sound intensity of4.7101watts per square meter?
  1. A radiation safety officer is working with112grams of a radioactive substance. After17days, the sample has decayed to80grams. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest day, what is the half-life of this substance?
Show Solution

f(t)=112e.019792t; half-life: about35 days

  1. Write the formula found in the previous exercise as an equivalent equation with basee.Express the exponent to five significant digits.
  1. A bottle of soda with a temperature of71°Fahrenheit was taken off a shelf and placed in a refrigerator with an internal temperature of35° F.After ten minutes, the internal temperature of the soda was63° F.Use Newton’s Law of Cooling to write a formula that models this situation. To the nearest degree, what will the temperature of the soda be after one hour?
Show Solution

T(t)=36e0.025131t+35;T(60)43oF

    1. The population of a wildlife habitat is modeled by the equationP(t)=3601+6.2e0.35t, wheretis given in years. How many animals were originally transported to the habitat? How many years will it take before the habitat reaches half its capacity?

Enter the data from the Table in #33 into a graphing calculator and graph the resulting scatter plot. Determine whether the data from the table would likely represent a function that is linear, exponential, or logarithmic.

x f(x)
1 3
2 8.55
3 11.79
4 14.09
5 15.88
6 17.33
7 18.57
8 19.64
9 20.58
10 21.42
Show Solution

logarithmic

Scatter plot representing the data in #33.

  1. The population of a lake of fish is modeled by the logistic equationP(t)=16,1201+25e0.75t, wheretis time in years. To the nearest hundredth, how many years will it take the lake to reach80of its carrying capacity?

For the following exercises, use a graphing utility to create a scatter diagram of the data given in the table. Observe the shape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round values to five decimal places.

x f(x)
1 20
2 21.6
3 29.2
4 36.4
5 46.6
6 55.7
7 72.6
8 87.1
9 107.2
10 138.1
Show Solution

exponential;y=15.10062(1.24621)x

Scatter plot representing the data in #35.

x f(x)
3 13.98
4 17.84
5 20.01
6 22.7
7 24.1
8 26.15
9 27.37
10 28.38
11 29.97
12 31.07
13 31.43
x f(x)
0 2.2
0.5 2.9
1 3.9
1.5 4.8
2 6.4
3 9.3
4 12.3
5 15
6 16.2
7 17.3
8 17.9
Show Solution

logistic;y=18.416591+7.54644e0.68375x

Scatter plot representing the data in #37.

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