Question 1

Find the value. Give an approximation to four decimal places.
-$\ln \left( 6.9 \times e ^ { - 6 } \right)$
A) $- 4.0685$
B) $1.9319$
C) $1.9315$
D) $7.9315$

Accepted Answer

The expression simplifies to $\ln(6.9) + \ln(e^{-6})$. Since $\ln(e^{-6}) = -6$, the value is $\ln(6.9) - 6$. Approximating $\ln(6.9)$ gives a value close to 1.9315, so the total is approximately $1.9315 - 6 = -4.0685$.

Question 2

Solve the problem.
-Which of the following is the same as $3 \log ( 4 x )$ for $x > 0$ ?
A) $\log 12 + \log x$
B) $\log \left( 64 x ^ { 3 } \right)$
C) $\log 64 \cdot \log x ^ { 3 }$
D) $\log ( 12 x )$

Accepted Answer

Using the properties of logarithms, $3 \log(4x)$ can be rewritten as $\log((4x)^3)$, which simplifies to $\log(64x^3)$.

Question 3

Find the value. Give an approximation to four decimal places.
-$\ln 295$
A) $108.8561$
B) $2.4698$
C) $0.1753$
D) $5.6870$

Accepted Answer

The natural logarithm of 295 is approximately 5.6870 when calculated to four decimal places.

Question 4

Use the change of base rule to find the logarithm to four decimal places.
-$$\log _ { 6 } 61.70$$
A) 0.4347
B) 1.7903
C) 2.3007
D) 10.2833

Accepted Answer

The change of base formula is $\log_b a = \frac{\log_c a}{\log_c b}$. Using base 10, $\log_{6} 61.70 = \frac{\log_{10} 61.70}{\log_{10} 6} \approx \frac{1.7903}{0.7782} \approx 2.3007$.

Question 5

Use the change of base rule to find the logarithm to four decimal places.
-$$\log _ { \sqrt { 6 } } 193.7$$
A)0)1701
B) 2.9392
C) 5.8784
D) 0.3891 338)

Accepted Answer

The change of base formula is $\log_b a = \frac{\log_k a}{\log_k b}$. Using base 10, $\log_{\sqrt{6}} 193.7 = \frac{\log_{10} 193.7}{\log_{10} \sqrt{6}}$. Calculating this gives approximately 5.8784.

Question 6

Find the value. Give an approximation to four decimal places.
-$\ln 0.000814$
A) $3.0894$
B) $7.1136$
C) $- 3.0894$
D) $- 7.1136$

Accepted Answer

The natural logarithm of a number less than 1 is negative, and using a calculator, $\ln 0.000814$ approximates to $-7.1136$.

Question 7

Find the value. Give an approximation to four decimal places.
-$\ln 0.987$
A) $0.0057$
B) $- 0.0131$
C) $- 0.0057$
D) $0.0131$

Accepted Answer

The natural logarithm of a number less than 1 (but greater than 0) is negative, which eliminates options A and D. Using a calculator or a logarithm table to find $\ln 0.987$ gives a value close to $-0.0131$, making option B correct.

Question 8

Use the change of base rule to find the logarithm to four decimal places.
-$$\log _ { 4.2 } 3.2$$
A) 0.5051
B) 0.7619
C) 1.2338
D) 0.8105

Accepted Answer

The answer of Use the change of base rule to...

Question 9

Find the value. Give an approximation to four decimal places.
-$\ln 26,700,000$
A) $7.4265$
B) $5.5872$
C) $0.0583$
D) $17.1002$

Accepted Answer

The natural logarithm of 26,700,000 is approximately 17.1002.

Question 10

Use the change of base rule to find the logarithm to four decimal places.
-$$\log _ { 2.3 } 262$$
A) 113.9130
B) 6.6854
C) 0.1496
D) 2.4183

Accepted Answer

The change of base formula is $\log_b a = \frac{\log_c a}{\log_c b}$, where $c$ can be any positive value different from 1. Using the common base of 10, the calculation is $\log_{2.3} 262 = \frac{\log_{10} 262}{\log_{10} 2.3}$. Calculating this gives approximately 6.6854.

Question 11

Find the value. Give an approximation to four decimal places.
-$\ln 797 + \ln 27$
A) $4.3343$
B) $4.1530$
C) $22.0190$
D) $9.9767$

Accepted Answer

The properties of logarithms allow us to combine the two logarithms into one by multiplication, since $\ln a + \ln b = \ln(ab)$. Thus, $\ln 797 + \ln 27 = \ln(797 \times 27)$. Calculating $797 \times 27$ gives 21519, and $\ln(21519)$ approximately equals $9.9767$.

Question 12

Find the value. Give an approximation to four decimal places.
-$$\ln \left( \frac { 98 } { 18 } \right)$$
A) $7.4753$
B) $1.6946$
C) $1.5863$
D) $- 3.4887$

Accepted Answer

To find the value of $\ln \left( \frac { 98 } { 18 } \right)$, first simplify the fraction: $\frac { 98 } { 18 } = \frac { 49 } { 9 }$. Then, use a calculator to find the natural logarithm of $\frac { 49 } { 9 }$, which gives an approximation of $1.6946$.

Question 13

Find the value. Give an approximation to four decimal places.
-Quail are game birds that fare poorly when their habitat is encroached upon. Wildlife biologists hav discovered that the population $\mathrm { P }$ of quail in a region is related to the percent of the region that has $\mathrm { b } _ { \mathrm { t } }$ paved with roads and parking lots, according to the formula
$$P = k \ln \left( \frac { 32 } { x + 4 } \right) , 0 \leq x \leq 28 \text {, }$$
where $x$ is the percent of the region that has been paved. For a particular rural region, $P = 1400$ wher $x = 0$. Predict what the quail population will be in this region when it becomes $6 \%$ paved. Round your answer to the nearest whole number.
A) 836 quail
B) 0 quail
C) 783 quail
D) 794 quail

Accepted Answer

The answer of Find the value. Give an approximation to...

Question 14

Find the value. Give an approximation to four decimal places.
-The logarithmic function $f ( x ) = - 100 + 87 \ln x$ models the number of visitors (in millions) to a certain country's museums, where $x$ is the number of years since the initial recording of the number of visitors. Use this function to estimate the number of visitors in year 60 . Round to the nearest tenth of a year.
A) $384.3$ million
B) $256.2$ million
C) $192.2$ million
D) $128.1$ million

Accepted Answer

The answer of Find the value. Give an approximation to...

Question 15

Find the value. Give an approximation to four decimal places.
-$\ln ( 454 \times 28 )$
A) $20.3868$
B) $4.1406$
C) $9.4503$
D) $3.8452$

Accepted Answer

The value can be found by first multiplying 454 by 28 to get 12712, and then taking the natural logarithm of 12712. Using a calculator, $\ln(12712) \approx 9.4503$.

Question 16

Use the change of base rule to find the logarithm to four decimal places.
-$$\log _ { 3 } 0.68$$
A) -0.3510
B) -0.7020
C) 0.3510
D) -2.8486

Accepted Answer

The change of base formula is $\log_b a = \frac{\log_c a}{\log_c b}$, where $c$ can be any positive value different from 1. Using the common logarithm (base 10), the calculation is $\log_3 0.68 = \frac{\log 0.68}{\log 3}$. Plugging these values into a calculator gives approximately -0.3510.

Question 17

Use the change of base rule to find the logarithm to four decimal places.
-$$\log _ { 9 } 0.912$$
A) -23.8530
B) -0.0419
C) -0.0400
D) 9.8684

Accepted Answer

The change of base formula is $\log_b a = \frac{\log_c a}{\log_c b}$, where $c$ can be any positive value different from 1. Using the common logarithm (base 10), the calculation is $\log_9 0.912 = \frac{\log 0.912}{\log 9}$. Calculating this gives approximately -0.0419.

Question 18

Find the value. Give an approximation to four decimal places.
-$$\ln \sqrt [ 5 ] { \mathrm { e } }$$
A) 4
B) $\frac { 1 } { 5 }$
C) 5
D) 6

Accepted Answer

The expression can be simplified using logarithmic and exponential properties. The natural logarithm of e, $\ln(e)$, is 1. Since the e is raised to the power of $\frac{1}{5}$ inside the logarithm, it simplifies to $\frac{1}{5}$.

Question 19

Use the change of base rule to find the logarithm to four decimal places.
-$$\log _ { 3 } 5$$
A) 2.9300
B) 1.4650
C) 0.6826
D) -1.4650

Accepted Answer

The answer of Use the change of base rule to...

Question 20

Find the value. Give an approximation to four decimal places.
-$\ln 131 - \ln 27$
A) $- 5.0939$
B) $8.1710$
C) $1.5794$
D) $1.4792$

Accepted Answer

Using the properties of logarithms, specifically the quotient rule $\ln a - \ln b = \ln(\frac{a}{b})$, we can simplify the expression to $\ln(\frac{131}{27})$. Calculating this value gives approximately 1.5794.

Quiz 5: Inverse, Exponential, and Logarithmic Functions

Find the value. Give an approximation to four decimal places. - $\ln \left( 6.9 \times e ^ { - 6 } \right)$

Solve the problem. -Which of the following is the same as $3 \log ( 4 x )$ for $x > 0$ ?

Find the value. Give an approximation to four decimal places. - $\ln 295$

Use the change of base rule to find the logarithm to four decimal places. - $\log _ { 6 } 61.70$

Use the change of base rule to find the logarithm to four decimal places. - $\log _ { \sqrt { 6 } } 193.7$

Find the value. Give an approximation to four decimal places. - $\ln 0.000814$

Find the value. Give an approximation to four decimal places. - $\ln 0.987$

Use the change of base rule to find the logarithm to four decimal places. - $\log _ { 4.2 } 3.2$

Find the value. Give an approximation to four decimal places. - $\ln 26,700,000$

Use the change of base rule to find the logarithm to four decimal places. - $\log _ { 2.3 } 262$

Find the value. Give an approximation to four decimal places. - $\ln 797 + \ln 27$

Find the value. Give an approximation to four decimal places. - $\ln \left( \frac { 98 } { 18 } \right)$

Find the value. Give an approximation to four decimal places. - $\ln ( 454 \times 28 )$

Use the change of base rule to find the logarithm to four decimal places. - $\log _ { 3 } 0.68$

Use the change of base rule to find the logarithm to four decimal places. - $\log _ { 9 } 0.912$

Find the value. Give an approximation to four decimal places. - $\ln \sqrt [ 5 ] { \mathrm { e } }$

Use the change of base rule to find the logarithm to four decimal places. - $\log _ { 3 } 5$

Find the value. Give an approximation to four decimal places. - $\ln 131 - \ln 27$

Quiz 5: Inverse, Exponential, and Logarithmic Functions

Find the value. Give an approximation to four decimal places. - ln⁡(6.9×e−6) \ln \left( 6.9 \times e ^ { - 6 } \right) ln(6.9×e−6)

Solve the problem. -Which of the following is the same as 3log⁡(4x) 3 \log ( 4 x ) 3log(4x) for x>0x > 0x>0 ?

Find the value. Give an approximation to four decimal places. - ln⁡295\ln 295ln295

Use the change of base rule to find the logarithm to four decimal places. - log⁡661.70\log _ { 6 } 61.70log6​61.70

Use the change of base rule to find the logarithm to four decimal places. - log⁡6193.7\log _ { \sqrt { 6 } } 193.7log6​​193.7

Find the value. Give an approximation to four decimal places. - ln⁡0.000814\ln 0.000814ln0.000814

Find the value. Give an approximation to four decimal places. - ln⁡0.987\ln 0.987ln0.987

Use the change of base rule to find the logarithm to four decimal places. - log⁡4.23.2\log _ { 4.2 } 3.2log4.2​3.2

Find the value. Give an approximation to four decimal places. - ln⁡26,700,000\ln 26,700,000ln26,700,000

Use the change of base rule to find the logarithm to four decimal places. - log⁡2.3262\log _ { 2.3 } 262log2.3​262

Find the value. Give an approximation to four decimal places. - ln⁡797+ln⁡27\ln 797 + \ln 27ln797+ln27

Find the value. Give an approximation to four decimal places. - ln⁡(9818) \ln \left( \frac { 98 } { 18 } \right) ln(1898​)

Find the value. Give an approximation to four decimal places. - ln⁡(454×28) \ln ( 454 \times 28 ) ln(454×28)

Use the change of base rule to find the logarithm to four decimal places. - log⁡30.68\log _ { 3 } 0.68log3​0.68

Use the change of base rule to find the logarithm to four decimal places. - log⁡90.912\log _ { 9 } 0.912log9​0.912

Find the value. Give an approximation to four decimal places. - ln⁡e5\ln \sqrt [ 5 ] { \mathrm { e } }ln5e​

Use the change of base rule to find the logarithm to four decimal places. - log⁡35\log _ { 3 } 5log3​5

Find the value. Give an approximation to four decimal places. - ln⁡131−ln⁡27\ln 131 - \ln 27ln131−ln27