Chapter 5

In Exercises 1–6, evaluate the function. If the value is not a rational number, round your answer to three decimal places. $$ \begin{array}{l}{\text { (a) } \sinh 3} \\ {\text { (b) } \tanh (-2)}\end{array} $$

Finding an Indefinite Integral In Exercises \(1-20\) , find the indefinite integral. $$ \int \frac{d x}{\sqrt{9-x^{2}}} $$

Evaluating a Logarithmic Expression In Exercises \(1-4\) , evaluate the expression without using a calculator. $$ \log _{2} \frac{1}{8} $$

Solving an Exponential or Logarithmic Equation In Exercises 1-16, solve for \(x\) accurate to three decimal places. $$ e^{\ln x}=4 $$

Evaluating a Logarithm In Exercises \(1-4,\) use a graphing utility to evaluate the logarithm by (a) using the natural logarithm key and (b) using the integration capabilities to evaluate the integral \(\int_{1}^{x}(1 / t) d t .\) $$ \ln 45 $$

Finding an Indefinite Integral In Exercises \(1-26,\) find the indefinite integral.. $$ \int \frac{5}{x} d x $$

Show that \(f\) and \(g\) are inverse functions (a) analytically and (b) graphically. \(f(x)=5 x+1, \quad g(x)=\frac{x-1}{5}\)

In Exercises 1–6, evaluate the function. If the value is not a rational number, round your answer to three decimal places. $$ \begin{array}{l}{\text { (a) } \cosh 0} \\ {\text { (b) } \operatorname{sech} 1}\end{array} $$

Finding an Indefinite Integral In Exercises \(1-20\) , find the indefinite integral. $$ \int \frac{d x}{\sqrt{1-4 x^{2}}} $$

Evaluating a Logarithmic Expression In Exercises \(1-4\) , evaluate the expression without using a calculator. $$ \log _{27} 9 $$

Solving an Exponential or Logarithmic Equation In Exercises 1-16, solve for \(x\) accurate to three decimal places. $$ e^{\ln 3 x}=24 $$

Evaluating a Logarithm In Exercises \(1-4,\) use a graphing utility to evaluate the logarithm by (a) using the natural logarithm key and (b) using the integration capabilities to evaluate the integral \(\int_{1}^{x}(1 / t) d t .\) $$ \ln 8.3 $$

Finding an Indefinite Integral In Exercises \(1-26,\) find the indefinite integral.. $$ \int \frac{10}{x} d x $$

Show that \(f\) and \(g\) are inverse functions (a) analytically and (b) graphically. \(f(x)=3-4 x, \quad g(x)=\frac{3-x}{4}\)

In Exercises 1–6, evaluate the function. If the value is not a rational number, round your answer to three decimal places. $$ \begin{array}{l}{\text { (a) } \operatorname{csch}(\ln 2)} \\ {\text { (b) } \operatorname{coth}(\ln 5)}\end{array} $$

Finding an Indefinite Integral In Exercises \(1-20\) , find the indefinite integral. $$ \int \frac{1}{x \sqrt{4 x^{2}-1}} d x $$

Evaluate the expression without using a calculator. \(\arcsin 0\)

Evaluating a Logarithmic Expression In Exercises \(1-4\) , evaluate the expression without using a calculator. $$ \log _{7} 1 $$

Solving an Exponential or Logarithmic Equation In Exercises 1-16, solve for \(x\) accurate to three decimal places. $$ e^{x}=12 $$

Evaluating a Logarithm In Exercises \(1-4,\) use a graphing utility to evaluate the logarithm by (a) using the natural logarithm key and (b) using the integration capabilities to evaluate the integral \(\int_{1}^{x}(1 / t) d t .\) $$ \ln 0.8 $$

Finding an Indefinite Integral In Exercises \(1-26,\) find the indefinite integral.. $$ \int \frac{1}{x+1} d x $$

Show that \(f\) and \(g\) are inverse functions (a) analytically and (b) graphically. \(f(x)=x^{3}, \quad \quad g(x)=\sqrt[3]{x}\)

In Exercises 1–6, evaluate the function. If the value is not a rational number, round your answer to three decimal places. $$ \begin{array}{l}{\text { (a) } \sinh ^{-1} 0} \\ {\text { (b) } \tanh ^{-1} 0}\end{array} $$

Finding an Indefinite Integral In Exercises \(1-20\) , find the indefinite integral. $$ \int \frac{12}{1+9 x^{2}} d x $$

Evaluate the expression without using a calculator. \(\arcsin \frac{1}{2}\)

Evaluating a Logarithmic Expression In Exercises \(1-4\) , evaluate the expression without using a calculator. $$ \log _{a} \frac{1}{a} $$

Solving an Exponential or Logarithmic Equation In Exercises 1-16, solve for \(x\) accurate to three decimal places. $$ 5 e^{x}=36 $$

Evaluating a Logarithm In Exercises \(1-4,\) use a graphing utility to evaluate the logarithm by (a) using the natural logarithm key and (b) using the integration capabilities to evaluate the integral \(\int_{1}^{x}(1 / t) d t .\) $$ \ln 0.6 $$

Finding an Indefinite Integral In Exercises \(1-26,\) find the indefinite integral.. $$ \int \frac{1}{x-5} d x $$

Show that \(f\) and \(g\) are inverse functions (a) analytically and (b) graphically. \(f(x)=1-x^{3}, \quad g(x)=\sqrt[3]{1-x}\)

In Exercises 1–6, evaluate the function. If the value is not a rational number, round your answer to three decimal places. $$ \begin{array}{l}{\text { (a) } \cosh ^{-1} 2} \\ {\text { (b) } \operatorname{sech}^{-1} \frac{2}{3}}\end{array} $$

Finding an Indefinite Integral In Exercises \(1-20\) , find the indefinite integral. $$ \int \frac{1}{\sqrt{1-(x+1)^{2}}} d x $$

Exponential and Logarithmic Forms of Equations In Exercises \(5-8,\) write the exponential equation as a logarithmic equation or vice versa. $$ \begin{array}{l}{\text { (a) } 2^{3}=8} \\ {\text { (b) } 3^{-1}=\frac{1}{3}}\end{array} $$

Evaluate the expression without using a calculator. \(\arccos \frac{1}{2}\)

Solving an Exponential or Logarithmic Equation In Exercises 1-16, solve for \(x\) accurate to three decimal places. $$ 9-2 e^{x}=7 $$

Finding an Indefinite Integral In Exercises \(1-26,\) find the indefinite integral.. $$ \int \frac{1}{2 x+5} d x $$

In Exercises 1–6, evaluate the function. If the value is not a rational number, round your answer to three decimal places. $$ \begin{array}{l}{\text { (a) } \operatorname{csch}^{-1} 2} \\ {\text { (b) } \operatorname{coth}^{-1} 3}\end{array} $$

Finding an Indefinite Integral In Exercises \(1-20\) , find the indefinite integral. $$ \int \frac{1}{4+(x-3)^{2}} d x $$

Exponential and Logarithmic Forms of Equations In Exercises \(5-8,\) write the exponential equation as a logarithmic equation or vice versa. $$ \begin{array}{l}{\text { (a) } 27^{2 / 3}=9} \\ {\text { (b) } 16^{3 / 4}=8}\end{array} $$

Evaluate the expression without using a calculator. \(\arccos 1\)

Solving an Exponential or Logarithmic Equation In Exercises 1-16, solve for \(x\) accurate to three decimal places. $$ 8 e^{x}-12=7 $$

Finding an Indefinite Integral In Exercises \(1-26,\) find the indefinite integral.. $$ \int \frac{9}{5-4 x} d x $$

Show that \(f\) and \(g\) are inverse functions (a) analytically and (b) graphically. \(f(x)=16-x^{2}, \quad x \geq 0, \quad g(x)=\sqrt{16-x}\)

In Exercises 7–14, verify the identity. $$ \tanh ^{2} x+\operatorname{sech}^{2} x=1 $$

Finding an Indefinite Integral In Exercises \(1-20\) , find the indefinite integral. $$ \int \frac{t}{\sqrt{1-t^{4}}} d t $$

Exponential and Logarithmic Forms of Equations In Exercises \(5-8,\) write the exponential equation as a logarithmic equation or vice versa. $$ \begin{array}{l}{\text { (a) } \log _{10} 0.01=-2} \\ {\text { (b) } \log _{0.5} 8=-3}\end{array} $$

Evaluate the expression without using a calculator. \(\arctan \frac{\sqrt{3}}{3}\)

Solving an Exponential or Logarithmic Equation In Exercises 1-16, solve for \(x\) accurate to three decimal places. $$ 50 e^{-x}=30 $$

Finding an Indefinite Integral In Exercises \(1-26,\) find the indefinite integral.. $$ \int \frac{x}{x^{2}-3} d x $$

Show that \(f\) and \(g\) are inverse functions (a) analytically and (b) graphically. \(f(x)=\frac{1}{x}, \quad \quad \quad g(x)=\frac{1}{x}\)