Chapter 5

Exer. 5-16: Determine whether the function \(f\) is one-to-one. $$ f(x)=|x| $$

Sketch the graph of \(f\) if \(a=2\). (a) \(f(x)=a^{x}\) (b) \(f(x)=-a^{x}\) (c) \(f(x)=3 a^{x}\) (d) \(f(x)=a^{x+3}\) (e) \(f(x)=a^{x}+3\) (f) \(f(x)=a^{x-3}\) (g) \(f(x)=a^{x}-3\) (h) \(f(x)=a^{-x}\) (i) \(f(x)=\left(\frac{1}{a}\right)^{x}\) (j) \(f(x)=a^{3-x}\)

Write the expression as one logarithm. $$ 5 \log _{a} x-\frac{1}{2} \log _{a}(3 x-4)-3 \log _{a}(5 x+1) $$

$$ e^{3 x}=e^{2 x-1} $$

Exer. 11-24: Find the exact solution, using common logarithms, and a two- decimal-place approximation of each solution, when appropriate. $$ 4^{2 x+3}=5^{x-2} $$

Exer. 11-12: Change to logarithmic form. (a) \(10^{4}=10,000\) (b) \(10^{-2}=0.01\) (c) \(10^{x}=38 z\) (d) \(e^{4}=D\) (e) \(e^{0.1 t}=x+2\)

Exer. 5-16: Determine whether the function \(f\) is one-to-one. $$ f(x)=3 $$

Write the expression as one logarithm. $$ \log \left(x^{3} y^{2}\right)-2 \log x \sqrt[3]{y}-3 \log \left(\frac{x}{y}\right) $$

Find the zeros of \(f\). $$ f(x)=x e^{x}+e^{x} $$

Exer. 11-24: Find the exact solution, using common logarithms, and a two- decimal-place approximation of each solution, when appropriate. $$ 2^{2 x-3}=5^{x-2} $$

Exer. 5-16: Determine whether the function \(f\) is one-to-one. $$ f(x)=\sqrt{4-x^{2}} $$

Sketch the graph of \(f\). $$f(x)=\left(\frac{2}{3}\right)^{-x}$$

Write the expression as one logarithm. $$ 2 \log \frac{y^{3}}{x}-3 \log y+\frac{1}{2} \log x^{4} y^{2} $$

Find the zeros of \(f\). $$ f(x)=-x^{2} e^{-x}+2 x e^{-x} $$

Exer. 11-24: Find the exact solution, using common logarithms, and a two- decimal-place approximation of each solution, when appropriate. $$ 3^{2-3 x}=4^{2 x+1} $$

Exer. 13-14: Change to exponential form. (a) \(\log x=-8\) (b) \(\log x=y-2\) (c) \(\ln x=\frac{1}{2}\) (d) \(\ln z=7+x\) (e) \(\ln (t-5)=1.2\)

Exer. 5-16: Determine whether the function \(f\) is one-to-one. $$ f(x)=2 x^{3}-4 $$

Sketch the graph of \(f\). $$f(x)=\left(\frac{2}{3}\right)^{x}$$

Write the expression as one logarithm. $$ \ln y^{3}+\frac{1}{3} \ln \left(x^{3} y^{6}\right)-5 \ln y $$

Find the zeros of \(f\). $$ f(x)=x^{3}\left(4 e^{4 x}\right)+3 x^{2} e^{4 x} $$

Exer. 11-24: Find the exact solution, using common logarithms, and a two- decimal-place approximation of each solution, when appropriate. $$ 2^{-x}=8 $$

Exer. 15-16: Find the number, if possible. (a) \(\log _{5} 1\) (b) \(\log _{3} 3\) (c) \(\log _{4}(-2)\) (d) \(\log _{7} 7^{2}\) (e) \(3^{\log _{3} 8}\) (f) \(\log _{5} 125\) (g) \(\log _{4} \frac{1}{16}\)

Exer. 5-16: Determine whether the function \(f\) is one-to-one. $$ f(x)=\frac{1}{x} $$

Sketch the graph of \(f\). $$f(x)=5\left(\frac{1}{2}\right)^{x}+3$$

Write the expression as one logarithm. $$ 2 \ln x-4 \ln (1 / y)-3 \ln (x y) $$

Find the zeros of \(f\). $$ f(x)=x^{2}\left(2 e^{2 x}\right)+2 x e^{2 x}+e^{2 x}+2 x e^{2 x} $$

Exer. 11-24: Find the exact solution, using common logarithms, and a two- decimal-place approximation of each solution, when appropriate. $$ 2^{-x^{2}}=5 $$

Exer. 15-16: Find the number, if possible. (a) \(\log _{8} 1\) (b) \(\log _{9} 9\) (c) \(\log _{5} 0\) (d) \(\log _{6} 6^{7}\) (e) \(5^{\log _{5} 4}\) (f) \(\log _{3} 243\) (g) \(\log _{2} 128\)

Exer. 5-16: Determine whether the function \(f\) is one-to-one. $$ f(x)=\frac{1}{x^{2}} $$

Sketch the graph of \(f\). $$f(x)=8(4)^{-x}-2$$

Solve the equation. $$ \log _{6}(2 x-3)=\log _{6} 12-\log _{6} 3 $$

Simplify the expression.$$ \frac{\left(e^{x}+e^{-x}\right)\left(e^{x}+e^{-x}\right)-\left(e^{x}-e^{-x}\right)\left(e^{x}-e^{-x}\right)}{\left(e^{x}+e^{-x}\right)^{2}} $$

Exer. 11-24: Find the exact solution, using common logarithms, and a two- decimal-place approximation of each solution, when appropriate. $$ \log x=1-\log (x-3) $$

Exer. 17-18: Find the number. (a) \(10^{\log 3}\) (b) \(\log 10^{5}\) (c) \(\log 100\) (d) \(\log 0.0001\) (e) \(e^{\ln 2}\) (f) \(\ln e^{-3}\) (g) \(e^{2+\ln 3}\)

Sketch the graph of \(f\). $$f(x)=-\left(\frac{1}{2}\right)^{x}+4$$

Solve the equation. $$ \log _{4}(3 x+2)=\log _{4} 5+\log _{4} 3 $$

Simplify the expression. \(\frac{\left(e^{x}-e^{-x}\right)^{2}-\left(e^{x}+e^{-x}\right)^{2}}{\left(e^{x}+e^{-x}\right)^{2}}\)

Exer. 11-24: Find the exact solution, using common logarithms, and a two- decimal-place approximation of each solution, when appropriate. $$ \log (5 x+1)=2+\log (2 x-3) $$

Exer. 17-18: Find the number. (a) \(10^{\log 7}\) (b) \(\log 10^{-6}\) (c) \(\log 100,000\) (d) \(\log 0.001\) (e) \(e^{\ln 8}\) (f) \(\ln e^{2 / 3}\) (g) \(e^{1+\ln 5}\)

Exer. 17-20: Use the theorem on inverse functions to prove that \(f\) and \(g\) are inverse functions of each other, and sketch the graphs of \(f\) and \(g\) on the same coordinate plane. $$ f(x)=x^{2}+5, x \leq 0 ; \quad g(x)=-\sqrt{x-5}, x \geq 5 $$

Sketch the graph of \(f\). $$f(x)=-3^{x}+9$$

Solve the equation. $$ 2 \log _{3} x=3 \log _{3} 5 $$

Exer. 11-24: Find the exact solution, using common logarithms, and a two- decimal-place approximation of each solution, when appropriate. $$ \log \left(x^{2}+4\right)-\log (x+2)=2+\log (x-2) $$

Exer. 19-34: Solve the equation. $$ \log _{4} x=\log _{4}(8-x) $$

Exer. 17-20: Use the theorem on inverse functions to prove that \(f\) and \(g\) are inverse functions of each other, and sketch the graphs of \(f\) and \(g\) on the same coordinate plane. $$ f(x)=-x^{2}+3, x \geq 0 ; \quad g(x)=\sqrt{3-x}, x \leq 3 $$

Solve the equation. $$ 3 \log _{2} x=2 \log _{2} 3 $$

Exer. 11-24: Find the exact solution, using common logarithms, and a two- decimal-place approximation of each solution, when appropriate. $$ \log (x-4)-\log (3 x-10)=\log (1 / x) $$

Exer. 19-34: Solve the equation. $$ \log _{3}(x+4)=\log _{3}(1-x) $$

Exer. 17-20: Use the theorem on inverse functions to prove that \(f\) and \(g\) are inverse functions of each other, and sketch the graphs of \(f\) and \(g\) on the same coordinate plane. $$ f(x)=x^{3}-4 ; \quad g(x)=\sqrt[3]{x+4} $$

Sketch the graph of \(f\). $$f(x)=2^{-|x|}$$