1.5.1: Exercises 1.5

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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: \(\displaystyle x=\frac{e^2-5}{2}\)