1.5.1: Exercises 1.5

Terms and Concepts

Exercise $\PageIndex{1}$

Explain the relationship between logarithmic functions and exponential functions.

Answer

For the same base, they are inverses of each other.

Exercise $\PageIndex{2}$

What questions are you answering when you evaluate $\log_5{(25)}$?

Answer

$25$ is $5$ raised to what power?

Exercise $\PageIndex{3}$

What is the value of the base for $\ln{(x)}$?

Answer

$e$

Exercise $\PageIndex{4}$

Explain why logarithms help solve exponential statements.

Answer

Logarithms help solve exponential statements because logarithms and exponentials are inverse functions.

Problems

Evaluate the given statement in exercises $\PageIndex{5}$ – $\PageIndex{8}$.

Exercise $\PageIndex{5}$

$\displaystyle \log_3{(81)}$

Answer

$4$

Exercise $\PageIndex{6}$

$\displaystyle \ln{(e^{5.7})}$

Answer

$5.7$

Exercise $\PageIndex{7}$

$\displaystyle e^{-\ln{(x})}$

Answer

$\displaystyle \frac{1}{x}$

Exercise $\PageIndex{8}$

$\displaystyle 4^{\log_2{(2^2)}}$

Answer

$\displaystyle 16$

Write the given statement as a single simplified logarithm in exercises $\PageIndex{9}$ – $\PageIndex{12}$.

Exercise $\PageIndex{9}$

$\displaystyle 4 \log_3{(2x)}-\log_3{(y^2)}$

Answer

$\displaystyle \log_3{\Big( \frac{16x^4}{y^2}\Big)}$

Exercise $\PageIndex{10}$

$\displaystyle \frac{2}{3} \ln{(x)} + 3\ln{(2y)}$

Answer

$\displaystyle \ln{(8x^{2/3}y^3)}$

Exercise $\PageIndex{11}$

$\displaystyle (2x)\log_2{(3)}+ \log_2{(5)}$

Answer

$\displaystyle \log_2{(5 (3^{2x}))}$

Exercise $\PageIndex{12}$

$\displaystyle 3\ln{(xy)}-2\ln{(x^2y)}$

Answer

$\displaystyle \ln{\Big( \frac{y}{x} \Big)}$

In exercises $\PageIndex{13}$ – $\PageIndex{17}$, solve the given problem for $x$, if possible. If a problem cannot be solved, explain why.

Exercise $\PageIndex{13}$

$\displaystyle 5^x = 25$

Answer

$\displaystyle x=2$

Exercise $\PageIndex{14}$

$\displaystyle 5^x = -5$

Answer

Not possible; we cannot raise a positive number to a power and get a negative number.

Exercise $\PageIndex{15}$

$\displaystyle 5^x = 0$

Answer

Not possible; we cannot raise a positive number to a power and get a zero.

Exercise $\PageIndex{16}$

$\displaystyle 5^x = 0.2$

Answer

$x=-1$

Exercise $\PageIndex{17}$

$\displaystyle 5^x = 1$

Answer

$x=0$

In exercises $\PageIndex{18}$ – $\PageIndex{25}$, solve the given problem for $x$.

Exercise $\PageIndex{18}$

$\displaystyle 3^{x-6} =2$

Answer

$x=6+\log_3{(2)}$

Exercise $\PageIndex{19}$

$\displaystyle 4^{2x-5} = 3$

Answer

$\displaystyle x=\frac{5+\log_4{(3)}}{2}$

Exercise $\PageIndex{20}$

$\displaystyle 2^{5x+6} = 4$

Answer

$\displaystyle x=\frac{-4}{5}$

Exercise $\PageIndex{21}$

$\displaystyle 6^{x+\pi} = 2$

Answer

$\displaystyle x=\log_6{(2)} - \pi$

Exercise $\PageIndex{22}$

$\displaystyle \bigg( \frac{1}{6}\bigg)^{-3x-2}=36^{x+1}$

Answer

$\displaystyle x=0$

Exercise $\PageIndex{23}$

$\displaystyle -15=-8\ln{(3x)}+7$

Answer

$\displaystyle x=\frac{1}{3} e^{11/4}$

Exercise $\PageIndex{24}$

$\displaystyle 2^x=3^{x-1}$

Answer

$\displaystyle x=-\frac{\ln{(3)}}{\ln{(2)} -\ln{(3)}}$

Exercise $\PageIndex{25}$

$\displaystyle 8= 4\ln{(2x+5)}$

Answer

$\displaystyle x=\frac{e^2-5}{2}$