Logarithmic and Exponential Functions Self Quiz--Solutions

1. Compute:

            a. log₆ 216

            b.
Solutions

a. log₆ 216 =log₆ 6³=3

b. = = = -2

If you had difficulty with this exercise, review the tutorial Logarithmic Functions.

2. Solve for x:

Solution

If you had difficulty with this exercise, review the tutorial Properties of Exponents.

3. Write as a single logarithm:

Solution

If you had difficulty with this exercise, review the tutorial Properties of Logarithms.

4. Solve for x: 2^3x+1= 5^x+6

Solution

2^3x+1=5^x+6

ln(2^3x+1)=ln (5^x+6)

(3x+1)ln(2)=(x+6)ln(5)

x(3ln(2)-ln(5))=6ln(5)-ln(2)

If you had difficulty with this exercise, review the tutorials Solving Exponential Equations and Solving Equations with Logarithms.

5. Write as a single exponential expression:

Solution

If you had difficulty with this exercise, review the tutorial Properties of Exponents.

6. What is the y-intercept of the graph of f(x)=4^x ?

Solution

When x=0, f(x)=4⁰=1. Thus the y-intercept of the graph of f(x)=4^x is (0,1).

If you had difficulty with this exercise, review the tutorial The Graph of an Exponential Function.

7. Given that log 2=.301 and log 5=.699, find

            a. log 8

            b. log 20

            c. log .2

Solution

            a. log 8=log 2³=3log 2=3(.301)=.903

            b. log 20= log((2²)(5))=log 2² +log 5=2log 2+ log 5=2(.301)+.699=1.301

            c. log .2=log 5^-1 = -1(log 5)= -.699

If you had difficulty with this exercise, review the tutorial Properties of Logarithms.

8. What is the y-intercept of the graph of g(x)=log₅x ? The x-intercept of the graph?

Solution

The function g(x)=log₅x is undefined at x=0 . The graph has no y-intercept. However, when log₅x=0, x=5⁰=1; thus the graph of the function has an x-intercept at (1,0).

If you had difficulty with this exercise, review the tutorial The Graph of a Logarithmic Function.

9. Which is larger, log₁₀5 or ln 5? Why?

Solution

Let log₁₀5=a and let ln 5=b.

By definition, 10^a=5 and e^b=5. Since 5<10, a<1; since e<5, b>1.

Thus log₁₀ 5=a<1<b=ln 5.

If you had difficulty with this exercise, review the tutorial Two Common Bases For Logarithms.

10. The population of a colony of bacteria doubles every 5 hours. If 1000 bacteria
are present initially, how many bacteria does the colony contain after 2
days?

Solution

At time t=0 hours, the colony contains 1000 bacteria. After t=5 hours, the colony contains 2(1000)=2000 bacteria. After t=10 hours, it doubles again; now the colony numbers 2(2(1000))=2²(1000) bacteria. Continuing, we see that in 2 days (or 48 hours), the population of the colony will have doubled times; thus after 2 days, there are bacteria in the colony.

If you had difficulty with this exercise, review the tutorial Real-World Applications of Exponential Functions.