Deck 5: Logarithmic Functions

Question 1

Let $f(x)=\log (4 x), g(x)=\log (2 x)$ , and $h(x)=\frac{f(x)-g(x)}{f(x)+g(x)}$ , with $x>0$ . Which of the following is a simplified formula for $h(x)$ ?

A) $\frac{\log (2 x)}{\log (6 x)}$
B) $\log \left(\frac{1}{3}\right)$
C) $2 \log (4 x)+2 \log (2 x)$
D) $\frac{\log (2)}{\log \left(8 x^{2}\right)}$

Answer 1

$\frac{\log (2)}{\log \left(8 x^{2}\right)}$

Question 2

The notation $\log _{b} x$ means that the $\log$ base is unspecified. Given that $\log _{b} x=3$ , $\log _{b} y=3.2$ , and $\log _{b} z=4.7$ , use properties of $\log$ to find $\log _{b}\left(\frac{x y^{3}}{z}\right)$ .

Answer 2

7.9

Question 3

Solve $58 e^{0.7 t}=7+t$ for $t$ to 2 decimal places if possible. If not possible, enter "not possible".

Answer 3

not possible

Question 4

Evaluate $\log (\sqrt[5]{10})$ without a calculator. Leave fractions in the form "A\B" (do not convert to decimals).

Question 5

Evaluate $e^{\ln \left(5^{2}\right)}$ without a calculator.

Question 6

Rewriting the statement $\ln \left(\frac{1}{8}\right)=-2.079$ using exponents gives

A) $e^{-2.079}=\frac{1}{8}$
B) $e^{-2.079}=8$
C) $e^{8}=\frac{1}{-2.079}$
D) $e^{8}=-2.079$

Question 7

Rewriting the statement $\log s=r$ using exponents gives

A) $10^{s}=r$
B) $10^{r}=s$
C) $10^{s}=1 / r$
D) $10^{r}=1 / s$

Question 8

Rewriting $10^{3}=1000$ using logs gives

A) $\log (1000)=-3$
B) $\log (3)=-1000$
C) $\log (1000)=3$
D) $\log (3)=1000$

Question 9

Rewriting $e^{6 a}=b$ using logs gives

A) $\ln a=b/6$
B) $\ln (b / 6)=a$
C) $\ln 6 a=b$
D) $\ln b=6 a$

Question 10

Solve $4^{x}=9$ using logs. Give your answer to 3 decimal places.

Question 11

Solve $\frac{1}{12}=\left(\frac{1}{6}\right)^{x}$ using logs. Give your answer to 3 decimal places.

Question 12

Does $\ln \left(x^{4} e^{-\ln \sqrt{x}}\right)=\frac{7}{2} \ln x$ ?

Question 13

Does $\frac{\log A^{4}-\log \sqrt{A}}{\log \left(e^{\ln e}\right)}=\frac{7}{2} \frac{\log A}{\log e}$ ?

Question 14

Let $n=\log p$ and $m=\log q$ . What is $\log \left(p^{5} q^{4}\right)$ ?

A) $5 n+4 m$
B) $20 \mathrm{~nm}$
C) $n^{5}+m^{4}$
D) $\mathrm{nm}^{9}$

Question 15

Let $n=\log p$ . What is $\log \sqrt[6]{p}$ ?

A) $\sqrt[6]{n}$
B) $\frac{1}{6} n$
C) $n^{6}$
D) $\frac{6}{n}$

Question 16

Let $m=\log q$ . What is $\log (100,000 q)$ ?

A) $100,000 m$
B) $5 m$
C) $5+m$
D) $100,000+m$

Question 17

Let $n=\log p$ and $m=\log q$ . What is $\log \frac{p^{6}}{q^{7}}$ ?

A) $\frac{n^{6}}{m^{7}}$
B) $\left(\frac{n}{m}\right)^{-1}$
C) $(n-m)^{-1}$
D) $6 n-7 m$

Question 18

Solve for $t: \log (t-500)=4$

Question 19

Solve for $t$ : $600 e^{0.7 t}=500 e^{0.1 t}$ . Round your answer to 3 decimal places.

Question 20

Solve $12 e^{2 t}-48=0$ for $t$ . Give your answer to 4 decimal places.

Question 21

Solve $\log x=2$ for $x$ .

Question 22

Evaluate $10^{\log 13}$

Question 23

Evaluate $\log \left(10^{19}\right)$

Question 24

Evaluate $e^{\ln 9}$

Question 25

Evaluate $\ln \left(e^{9}\right)$

Question 26

If the equation $Q=8 \cdot 3^{1.7 t}$ is converted to the form $Q=a e^{k t}$ , then $a=$ ----------- and $k=$ ------------- Give k to 4 decimal places.

Question 27

Let $y=20 e^{-0.07 t}$ , with $t$ measured in years. What is the continuous percent decay rate per year?

Question 28

Let $y=60 e^{-0.07 t}$ , with $t$ measured in years. What is the percent annual decay rate?
Round to 2 decimal places.

Question 29

The doubling time for a bank account that is growing by $2.6 \%$ per year (compounded continually) is---------- years. Round 1 decimal place.

Question 30

If $Q=\frac{1}{7} e^{0.062 t}$ is converted to the form $Q=a b^{t}$ , then $b=\ldots$ . Round to 3 decimal places.

Question 31

What is the half-life of a substance that decays at a rate of $3.21 \%$ per day? Round your answer to the nearest hundredth of a day.

Question 32

What annual interest rate, compounded monthly, is equivalent to an annual interest rate of 5.4% compounded continuously? Round your answer to the nearest thousandth of a percent, if necessary.

Question 33

You invest $\$ 4,000$ in an account earning $4 \%$ annual interest. At the same time, you invest $\$ 8,000$ in a different account earning $2 \%$ annual interest. In both accounts, the interest is compounded continuously. When do the accounts each hold the same amount? Round your answer to the nearest year.

Question 34

A small country is experiencing hyperinflation of $59 \%$ per month.

A) By what percent have prices climbed after 5 months?
B) If an item currently costs $\$ 12$ , how much will it cost after 1 year of such inflation?

Question 35

Suppose the population of an endangered species is cut in half every 60 years. Assume exponential decay.

A) What is the annual growth rate?
B) What is the continuous growth rate?
Give your answers correct to 4 decimal places.

Question 36

If $11 \%$ of a radioactive substance decays in 8 hours, what is the half-life of the substance in hours? Round your answer to 3 decimal places.

Question 37

If $11 \%$ of a radioactive substance decays in 6 hours, what percent of the substance is left after 11 hours? Round your answer to 3 decimal places, if necessary.

Question 38

The half-life of a substance is 34 hours. If there are initially 125 grams of the substance, write an exponential formula to determine the amount $Q$ remaining after $t$ hours.

Question 39

The half-life of a substance is 37 hours. If there are initially 145 grams of the substance, how many grams are remaining after 52 hours? Round your answer to 3 decimal places.

Question 40

What is the doubling time of a population growing by $12 \%$ per year? Round your answer to the nearest hundredth of a year.

Question 41

A town with an initial population of 14,000 people doubles every 40 years. Write an exponential equation to determine the town's population $P$ after $t$ years.

Question 42

A town with an initial population of 12,000 people doubles every 41 years. Determine the town's population after 12 years.

Question 43

What annual interest rate, compounded quarterly, is equivalent to an annual interest rate of $5.5 \%$ compounded continuously? Round your answer to the nearest thousandth of a percent, if necessary.

Question 44

A certain country is experiencing deflation and prices are falling by $2.9 \%$ each month. If a certain item currently costs $\$ 31$ , how much will the item cost in a year?

Question 45

Suppose a certain population decreases by one third every 4 years. If the population is initially 17,000 , what is the size of the population after 10 years? Round your answer to the nearest whole number.

Question 46

What is the domain of the function $y=1+\ln (4-x)$ ?

A) $(-\infty, 4)$
B) $(-\infty, 3)$
C) $(-\infty, 1)$
D) $(-\infty, \infty)$

Question 47

The following figure shows the graphs of
(A). $y=e^{-x}$
(B). $y=10^{-x}$
(C). $y=\ln (-x)$
(D). $y=\log (-x)$ .

Which one is the graph of A?

Question 48

The vertical intercept of the graph of $y=e^{x-4}$ is at $y=\ldots$ . If necessary, round to 2 decimal places.

Question 49

The vertical asymptote of the graph of $y=5+\ln (4-x)$ is at $x=$ ----------- Round to 2 decimal places if necessary.

Question 50

Let $f(x)=\log (x-1)-1$ .

A) $\operatorname{Graph} f$ .
B) What are the asymptotes?
C) What are the intercepts if they exist?

Question 51

What is the hydrogen ion concentration $\left[\mathrm{H}^{+}\right]$ for a substance with a $\mathrm{pH}$ of 6.3 ? Express your answer as a power of 10 .

Question 52

Sound A is $32 \mathrm{~dB}$ . Sound B is $54 \mathrm{~dB}$ . How many times more intense is Sound $B$ than Sound A? Round your answer to 2 decimal places if necessary.

Question 53

How many times more powerful is an earthquake measuring 5.6 on the Richter scale than an earthquake measuring 4.5 on the Richter scale? Round your answer to two decimal places.

Question 54

Find a possible formula for the following graph using either logarithms or exponentials.

Question 55

Find the domain of the function $f(x)=\sqrt{9 \ln (x-7)}-6$ .

Question 56

Find the domain of the function $f(x)=\sqrt{-5 \ln (x-5)}-6$ .

Question 57

Consider the function $f(x)=-55^{-x}+2$ .

A) Graph the function.
B) State the domain.
C) State all asymptotes.

Question 58

Find $\lim _{x \rightarrow 6^{+}} \ln (x-6)$ .

Question 59

Find $\lim _{x \rightarrow 5^{+}} \ln (5-x)$ .

Question 60

An earthquake measured 5.1 on the Richter scale. How many times more powerful were the seismic waves than standard seismic waves?

Question 61

Suppose earthquake A measured 3.6 on the Richter scale. What was the Richter scale measure of earthquake B if it was 1700 times more powerful than earthquake A?
Round your answer to 2 decimal places.

Question 62

What is the domain of $\sqrt{\left(e^{3 t}-7\right)^{2}-4}$ ?

Question 63

Find the domain and asymptotes of the function $f(x)=-4\left[\ln \left(x^{2}-4\right)-\ln (x+2)\right]$

Question 64

Find the domain of the function $f(x)=\log \left(x^{2}-16\right)-\log (-3 x)$ .

Question 65

The hydrogen ion concentration $\left[\mathrm{H}^{+}\right]$ for a certain substance is 0.000316 . What is the $\mathrm{pH}$ of the substance?

Question 66

What is the range of the function $y=\ln (x-6)$ ?

Question 67

What is the range of the function $y=\ln \left(\frac{x}{3}\right)$ ?

Question 68

By what order of magnitude do the numbers 0.000001 and $100,000,000$ differ?

Question 69

If you wished to graph the number of cancer cases in the US per year over the past 50 years on a standard sheet of paper, would you use a standard or a logarithmic scale?

Question 70

The following table gives $z$ as a linear function of $x$ .

If $z=\ln y$ , which of the following is a formula for $y$ in terms of $x$ ?

A) $y=\ln (-1+3 x)$
B) $y=e^{-1+3 x}$
C) $y=-1 \ln (3 x)$
D) $y=-1 e^{3 x}$

Question 71

The following table gives $z$ as a linear function of $x$ .

If $z=\ln y$ , which of the following is a formula for $y$ in terms of $x$ ?

A) $y=\ln (-2+-5 x)$
B) $y=e^{-2-5 x}$
C) $y=-2 \ln (-5 x)$
D) $y=-2 e^{-5 x}$

Question 72

The following table gives $z$ as a linear function of $x$ .

If $z=\ln y$ , find a formula for $y$ in terms of $x$ .

Question 73

The following table gives $z$ as a linear function of $x$ .

If $z=\ln u$ and $u=\ln y$ , find a formula for $y$ in terms of $x$ .

Question 74

The following table gives $w$ as a linear function of $u$ .

If $w=\ln v$ , find $v(u)$ at $u=10$ . Give your answer in exponential form.

Question 75

The following table gives $v$ as a linear function of $w$ .

If $w=\ln x$ and $v=\ln y$ , which of the following is a formula for $y$ in terms of $x$ ?

A) $y=3 x$
B) $y=\ln (3 x)$
C) $y=x^{3}$
D) $y=e^{3 x}$

Question 76

The following table gives $v$ as a linear function of $w$ .

If $w=\ln x$ and $v=\ln y$ , find a formula for $y$ in terms of $x$ .

Question 77

The following table gives $v$ as a linear function of $w$ .

If $w=\ln x$ and $v=\ln y$ , find $y(x)$ at $x=15$ . Give your answer in exponential form.

Question 78

The following table gives $v$ as a linear function of $w$ .

If $w=\ln x, v=\ln u$ , and $u=\ln y$ , find a formula for $y$ in terms of $x$ .

Question 79

The following table gives $v$ as a linear function of $w$ .

If $w=\ln x$ and $v=\ln y$ , find a formula for $y$ in terms of $x$ .

Question 80

Use linear regression on the values $t$ and $\ln r$ to fit a function of the form $\ln r=b+m t$ .

Use a calculator program to find the slope of the regression line for this data. Round to 4 decimal places.