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Mathematics LibreTexts

6.4E Exercises

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Definition True/False

Answer True/False. If False, explain why.

  1. If logax=y, then ax=y.
  2. If logba=c, then bc=a.
  3. There is a logarithmic function with base 1, f(x)=log1x.
  4. The natural log function lnx is the logarithmic function with base π.
  5. Let f(x)=logax be a logarithmic function (a>0,a1). The domain of f is {x|x0}.
  6. Let f(x)=logax be a logarithmic function (a>0,a1). The domain of f is {x|x>0}.
  7. I am allowed to plug in x=2 to the function f(x)=ln(x+3).
  8. I am allowed to plug in x=2 to the function f(x)=ln(x+1).
  9. For any base a, loga(a)=1.
  10. For any base a, loga(0)=1.
  11. For any base a, loga(1)=0.
  12. For f(x)=log10x and g(x)=10x, we know (gf)(x)=x for all x>0.
Answer
  1. F, this does not match the definition. It must be ay=x.
  2. T
  3. F, we do not allow the base a=1. (Why doesn't it make sense?)
  4. F, it's base e.
  5. F, we can't have because 0 is not allowed.
  6. T
  7. T
  8. F, if x=2 then the input to the log is 2+1=1, a negative number.
  9. T
  10. F, 0 isn't even in the domain.
  11. T
  12. T
Graphs True/False

Answer True/False. If False, justify or find the error.

  1. The graph of f(x)=log3x passes through (1,0).
  2. The graph of f(x)=logax passes through (1,a) for any base a.
  3. The graph of f(x)=logax passes through (a,1) for any base a.
  4. The graph of the function f(x)=log3x is the reflection of the graph of g(x)=log13x across the x-axis.
  5. The graph of the function f(x)=lnx is the reflection of the graph of g(x)=ex across the x-axis.
  6. The graph of f(x)=log2x exhibits faster growth than the graph of g(x)=lnx, for x>1.
  7. The graph of f(x)=log10x exhibits faster growth than the graph of g(x)=lnx, for x>1.
Answer
  1. T
  2. F, it passes through (1,0).
  3. T
  4. T
  5. F, it's the reflection across the line y=x because they're inverse functions.
  6. T
  7. F, the base 10>e.
Matching Graphs

By analyzing signal points like (a,1) and using your knowledge of logarithmic function graphs, match the graphs to their functions.

log6.png log4.png log1.png log5.png
1. 2. 3. 4.
f(x)=log10x g(x)=lnx h(x)=log12x p(x)=log2x
Answer
  1. p
  2. g
  3. f
  4. h
Working With Logarithms

Without using a calculator, compute the following using the definition and Log Laws:

  1. log39
  2. log17(1)
  3. log2(14)
  4. lne
  5. lne4
  6. log101000
  7. log24+2log28
  8. log3(6)log3(2)
Answer
  1. 2
  2. 0
  3. 2
  4. 1
  5. 4
  6. 3
  7. 8
  8. 1 (Hint: consider log3(6)=log3(23).)
Condensing/Expanding

Fully expand or fully condense. Simplify if reasonable.

  1. log2(x2)
  2. log10(100x2y3)
  3. ln(A2+2AB+B2)
  4. log311+log37
  5. lnx+lny2lnz3ln(w+1)
  6. 3loga(A+B)4loga(AB)
Answer
  1. log2x1
  2. 2+2log10x+3log10y
  3. 2ln(A+B)
  4. log3(77)
  5. ln(xyz2(w+1)3)
  6. loga((A+B)3(AB)4)
Logarithms Vs. Exponentials True/False

Answer True/False. Justify your answer.

  1. log2(2x)=x
  2. log2(8)=3
  3. For any base a, loga(xa)=x.
  4. elnx=x (for x>0)
  5. elog10x=x (for x>0)
  6. 10log10x=x (for x>0)
Answer
  1. T, inverse functions.
  2. T, write as log2(23)=3log22 or translate with definition.
  3. F, it should be loga(ax)=x.
  4. T, inverse functions.
  5. F, the bases must match.
  6. T, inverse functions.
Solving for x

By translating back and forth from logarithm form to exponential form, solve for x in the equations below.

  1. log10(x)=3
  2. log2(x)=1
  3. ln(x+1)=4
  4. log5(5x)=2
  5. ln(x)3=0
  6. 2lnx=4
Answer
  1. x=1000
  2. x=2
  3. x=e41
  4. x=5
  5. x=e3
  6. x=e2
Applications

1. We saw how exponential functions can be used to calculate interest compounding annually. In fact, you can choose to compound multiple times a year, or even continuously. For continuously compounding interest, the amount of money you have at time t years after investing an initial amount A0 is given by A=A0ert, where r is the interest rate written as a decimal. Say I want to double my money, so that A is 2A0. Then my equation becomes 2=ert. Give the equivalent logarithmic equation. Solve it for t. If the interest rate is 12%, find the time t it will take to double my money. (Use a calculator if needed.)

2. If I take a turkey out of the oven with internal temperature 165F and place it in an ambient room temperature of 65F, then the amount of time (in minutes) it will take to cool to an internal temperature of T is given by

t=40ln(T65100)

Use a calculator to find how long it takes to cool down to 150F, 125F, and 100F.

Answer

1. The equivalent equation is ln(2)=rt. Solved for t, we have t=ln2r. If r=.12, then t5.78 years.

2. For T=150F, t6.5 minutes. For T=125F, t20.4 minutes. For T=100F, t42 minutes.


This page titled 6.4E Exercises is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Lydia de Wolf.

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