Chapter 4

State whether each function given by a table is one-to-one. Explain your reasoning. $$\begin{array}{cc}x & f(x) \\\\-3 & 6 \\\\-2 & -8 \\\0 & 0 \\\1 & 8 \\\3 & -6\end{array}$$

In Exercises \(15-20,\) use the properties of logarithms to simplify each expression by eliminating all exponents and radicals. Assume that \(x, y > 0\). $$\log \frac{\sqrt[4]{x}}{y^{-1}}$$

Evaluate each expression without using a calculator. $$\ln e^{2}$$

Sketch the graph of each function. $$g(x)=\left(\frac{1}{4}\right)^{x}$$

Solve the exponential equation. Round to three decimal places, when needed. $$5^{x+5}=3^{-2 x+1}$$

Use \(f(x)=3 \ln x-4\). Evaluate \(f(1)\).

State cohether each function given by a table is one-to-one. Explain your reasoning. $$\begin{array}{cc}x & f(x) \\\\-3 & 4 \\\\-1 & 7 \\\0 & 4 \\\1 & 5 \\\3 & 12\end{array}$$

In Exercises \(15-20,\) use the properties of logarithms to simplify each expression by eliminating all exponents and radicals. Assume that \(x, y > 0\). $$\log \frac{\sqrt[3]{x}}{y^{2}}$$

Evaluate each expression without using a calculator. $$\ln \sqrt{e}$$

Sketch the graph of each function. $$g(x)=\left(\frac{1}{5}\right)^{x}$$

Solve the exponential equation. Round to three decimal places, when needed. $$1000 e^{0.04 x}=2000$$

Use \(f(x)=3 \ln x-4\). For what value of \(x\) will \(f(x)=2 ?\)

State whether each function given by a table is one-to-one. Explain your reasoning. $$\begin{array}{cc}x & f(x) \\\\-2 & -6 \\\\-1 & 5 \\\0 & 9 \\\1 & 4 \\\2 & 9\end{array}$$

In Exercises \(21-30,\) write each logarithm as a sum and\or difference of logarithmic expressions. Eliminate exponents and radicals and evaluate logarithms wherever possible. Assume that \(a, x, y\) \(z>0\) and \(a \neq 1\). $$\log \frac{x^{2} y^{5}}{10}$$

Evaluate each expression without using a calculator. $$\ln e^{1 / 3}$$

Sketch the graph of each function. $$f(x)=2(3)^{-x}$$

Solve the exponential equation. Round to three decimal places, when needed. $$250 e^{0.05 x}=400$$

Use \(f(x)=3 \ln x-4\). For what value of \(x\) will \(f(x)=3 ?\)

State whether each function given by a table is one-to-one. Explain your reasoning. $$\begin{aligned}&x \quad f(x)\\\&\begin{array}{cc}-2 & -9 \\\\-1 & -8\\\0&-7\\\1&-6\\\2&-5\end{array}\end{aligned}$$

In Exercises \(21-30,\) write each logarithm as a sum and\or difference of logarithmic expressions. Eliminate exponents and radicals and evaluate logarithms wherever possible. Assume that \(a, x, y\) \(z>0\) and \(a \neq 1\). $$\log \frac{x^{5} y^{4}}{1000}$$

Evaluate each expression without using a calculator. $$\ln \frac{1}{e}$$

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

Solve the exponential equation. Round to three decimal places, when needed. $$5 e^{x}+7=32$$

In Exercises \(21-30,\) write each logarithm as a sum and\or difference of logarithmic expressions. Eliminate exponents and radicals and evaluate logarithms wherever possible. Assume that \(a, x, y\) \(z>0\) and \(a \neq 1\). $$\ln \frac{\sqrt[3]{x^{2}}}{e^{2}}$$

Evaluate each expression without using a calculator. $$\log 10^{x+y}$$

Sketch the graph of each function. $$f(x)=2 e^{x}$$

Solve the exponential equation. Round to three decimal places, when needed. $$4 e^{x}+6=22$$

In Exercises \(21-30,\) write each logarithm as a sum and\or difference of logarithmic expressions. Eliminate exponents and radicals and evaluate logarithms wherever possible. Assume that \(a, x, y\) \(z>0\) and \(a \neq 1\). $$\ln \frac{\sqrt[4]{y^{3}}}{e^{5}}$$

Evaluate each expression without using a calculator. $$\ln e^{x-z}$$

Sketch the graph of each function. $$g(x)=5 e^{x}$$

Solve the exponential equation. Round to three decimal places, when needed. $$2\left(0.8^{x}\right)-3=8$$

In Exercises \(21-30,\) write each logarithm as a sum and\or difference of logarithmic expressions. Eliminate exponents and radicals and evaluate logarithms wherever possible. Assume that \(a, x, y\) \(z>0\) and \(a \neq 1\). $$\log _{a} \frac{\sqrt{x^{2}+y}}{a^{3}}$$

Evaluate each expression without using a calculator. $$\log 10^{k}$$

Sketch the graph of each function. $$f(x)=2+3 e^{x}$$

Solve the exponential equation. Round to three decimal places, when needed. $$4\left(1.2^{x}\right)-4=9$$

In Exercises \(21-30,\) write each logarithm as a sum and\or difference of logarithmic expressions. Eliminate exponents and radicals and evaluate logarithms wherever possible. Assume that \(a, x, y\) \(z>0\) and \(a \neq 1\). $$\log _{a} \frac{\sqrt{x^{3} y+1}}{a^{4}}$$

Evaluate each expression without using a calculator. $$\ln e^{w}$$

Sketch the graph of each function. $$f(x)=5+2 e^{x}$$

It takes 5700 years for an initial amount \(A_{0}\) of carbon- 14 to break down into half the amount, \(\frac{A_{0}}{2}\) (a) Given an initial amount of \(A_{0}\) grams of carbon- 14 at time \(t=0,\) find an exponential decay model, \(A(t)=A_{0} e^{i t},\) that gives the amount of carbon-14 at time \(t, t \geq 0\). (b) Calculate the time required for \(A_{0}\) grams of carbon- 14 to decay to \(\frac{1}{3} A_{0}\).

Solve the exponential equation. Round to three decimal places, when needed. $$e^{x^{2}+1}-2=3$$

In Exercises \(21-30,\) write each logarithm as a sum and\or difference of logarithmic expressions. Eliminate exponents and radicals and evaluate logarithms wherever possible. Assume that \(a, x, y\) \(z>0\) and \(a \neq 1\). $$\log _{a} \sqrt{\frac{x^{6}}{y^{3} z^{5}}}$$

Evaluate each expression without using a calculator. $$\log _{2} \sqrt{2}$$

Sketch the graph of each function. $$g(x)=10(2)^{x}$$

The half-life of plutonium-238 is 88 years. (a) Given an initial amount of \(A_{0}\) grams of plutonium238 at time \(t=0,\) find an exponential decay model, \(A(t)=A_{0} e^{k t},\) that gives the amount of plutonium238 at time \(t, t \geq 0\). (b) Calculate the time required for \(A_{0}\) grams of plutonium- 238 to decay to \(\frac{1}{3} A_{0}\).

Solve the exponential equation. Round to three decimal places, when needed. $$5+e^{x^{2}+1}=8$$

In Exercises \(21-30,\) write each logarithm as a sum and\or difference of logarithmic expressions. Eliminate exponents and radicals and evaluate logarithms wherever possible. Assume that \(a, x, y\) \(z>0\) and \(a \neq 1\). $$\log _{a} \sqrt{\frac{z^{5}}{x y^{4}}}$$

Evaluate each expression without using a calculator. $$\log _{7} 49$$

Sketch the graph of each function. $$h(x)=-5(3)^{x}$$

Solve the exponential equation. Round to three decimal places, when needed. $$9-e^{x^{2}-1}=2$$

State cohether each function is one-to-one. $$f(x)=-3 x+2$$