Chapter 5

Find the logarithm, without using a calculator. $$\log 10,000$$

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

State which of the following models might be appropriate for the given scatter plot of data (more than one model may be appropriate $$\begin{array}{|l|l|} \hline \m{}{|c|}\text { Model } & \text { Corresponding Function } \\ \hline \text { A. Linear } & y=a x+b \\ \hline \text { B. Quadratic } & y=a x^{2}+b x+c \\ \hline \text { C. Power } & y=a x^{r} \\ \hline \text { D. Cubic } & y=a x^{3}+b x^{2}+c x+d \\ \hline \text { E. Exponential } & y=a b^{x} \\ \hline \text { F. Logarithmic } & y=a+b \ln x \\ \hline \text { G. Logistic } & y=\frac{a}{1+b e^{-k x}} \\ \hline \end{array}$$ (GRAPH CAN NOT COPY)

Solve the equation without using logarithms. $$3^{x}=81$$

Simplify the expression. Assume \(a, b, c, d>0\) $$\left(25 k^{2}\right)^{3 / 2}\left(16 k^{1 / 3}\right)^{3 / 4}$$

Write the given expression as a single logarithm. $$\ln x^{2}+3 \ln y$$

Find the logarithm, without using a calculator. $$\log .001$$

Sketch a complete graph of the function. $$f(x)=(1.001)^{-x}$$

Solve the equation without using logarithms. $$5^{x}-2=23$$

Simplify the expression. Assume \(a, b, c, d>0\) $$\left(4 x^{5 / 6}\right)\left(2 y^{3 / 4}\right)\left(x^{7 / 6}\right)\left(3 y^{-1 / 4}\right)$$

Write the given expression as a single logarithm. $$-5(\ln x)+\ln 4 y-\ln 3 z$$

Find the logarithm, without using a calculator. $$\log \frac{\sqrt{10}}{1000}$$

Sketch a complete graph of the function. $$g(x)=(5 / 2)^{x}$$

Solve the equation without using logarithms. $$3^{x+1}=9^{5 x}$$

Simplify the expression. Assume \(a, b, c, d>0\) $$\left(c^{2 / 5} d^{-2 / 3}\right)\left(c^{6} d^{3}\right)^{4 / 3}$$

Write the given expression as a single logarithm. $$\log \left(x^{2}-9\right)-\log (x+3)$$

Sketch a complete graph of the function. $$g(x)=(1.001)^{x}$$

Simplify the expression. Assume \(a, b, c, d>0\) $$\left(\sqrt[3]{3} x^{2} y\right)\left(\sqrt[3]{9} x^{-1 / 3} y^{3 / 5}\right)^{-2}$$

Write the given expression as a single logarithm. $$3(\log 2 x)-4[\log x-\log (y-5)]$$

Solve the equation without using logarithms. $$3^{7 x}=9^{2 x-5}$$

Find the logarithm, without using a calculator. $$\log \sqrt[3]{.01}$$

Sketch a complete graph of the function. $$h(x)=(1 / \pi)^{x}$$

Write the given expression as a single logarithm. $$2(\ln x)-3\left(\ln x^{2}+\ln x\right)$$

Simplify the expression. Assume \(a, b, c, d>0\) $$\frac{\left(x^{2}\right)^{1 / 3}\left(y^{2}\right)^{2 / 3}}{3 x^{2 / 3} y^{2}}$$

Translate the given logarithmic statement into an equivalent exponential statement. $$\log 1000=3$$

Sketch a complete graph of the function. $$h(x)=(1 / e)^{-x}$$

Write the given expression as a single logarithm. $$-\log \left(\frac{3 \sqrt{x}}{2}\right)-\log (\sqrt{5 x})$$

Simplify the expression. Assume \(a, b, c, d>0\) $$\frac{\left(a^{1 / 2} b^{2}\right)^{3}\left(a^{1 / 2} b^{0} c\right)}{\left(a b^{2}\right)^{2}\left(b c^{5}\right)^{0}}$$

Solve the equation without using logarithms. $$7^{x^{2}+3 x}=1 / 49$$

Translate the given logarithmic statement into an equivalent exponential statement. $$\log .001=-3$$

Simplify the expression. Assume \(a, b, c, d>0\) $$\frac{(7 a)^{2}(5 b)^{3 / 2}}{(5 a)^{3 / 2}(7 b)^{4}}$$

Solve the equation without using logarithms. $$9^{x^{2}}=3^{-5 x-2}$$

Translate the given logarithmic statement into an equivalent exponential statement. $$\log 750=2.88$$

Sketch a complete graph of the function. $$f(x)=1-2^{-x}$$

Simplify the expression. Assume \(a, b, c, d>0\) $$\frac{\sqrt{a b} \sqrt[3]{a b^{4}}}{\sqrt{a}(\sqrt[3]{b})^{4}}$$

Solve the equation without using logarithms. $$5^{2 x^{2}+3 x}=25^{6-x}$$

Translate the given logarithmic statement into an equivalent exponential statement. $$\log (.8)=-.097$$

Sketch a complete graph of the function. $$g(x)=(1.2)^{x}+(.8)^{-x}$$

Simplify the expression. Assume \(a, b, c, d>0\) $$\left(a^{x^{2}}\right)^{1 / x}$$

Solve the equation. First express your answer in terms of natural logarithms (for instance, \(x=(2+\ln 5) /(\ln 3)) .\) Then use a calculator to find an approximation for the answer. $$3^{x}=5$$

Translate the given logarithmic statement into an equivalent exponential statement. $$\ln 3=1.0986$$

Sketch a complete graph of the function. $$h(x)=2^{x^{2}}$$

Solve the equation. First express your answer in terms of natural logarithms (for instance, \(x=(2+\ln 5) /(\ln 3)) .\) Then use a calculator to find an approximation for the answer. $$2^{x}=9$$

Sketch a complete graph of the function. $$h(x)=2^{-x^{2}}$$

Compute the ratios of successive entries in the table to determine whether or not an exponential model is appropriate for the data. $$\begin{array}{|l|l|l|l|l|l|l|} \hline x & 0 & 2 & 4 & 6 & 8 & 10 \\ \hline y & 3 & 15.2 & 76.9 & 389.2 & 1975.5 & 9975.8 \\ \hline \end{array}$$

Compute and simplify. $$x^{1 / 2}\left(x^{2 / 3}-x^{4 / 3}\right)$$

Solve the equation. First express your answer in terms of natural logarithms (for instance, \(x=(2+\ln 5) /(\ln 3)) .\) Then use a calculator to find an approximation for the answer. $$2^{x}=3^{x-1}$$

In Exercises \(11-16,\) let \(u=\ln x\) and \(v=\ln y .\) Write the given expression in terms of u and v. For example, $$\ln x^{3} y=\ln x^{3}+\ln y=3 \ln x+\ln y=3 u+v$$ $$\ln \left(x^{2} y^{5}\right)$$

Translate the given logarithmic statement into an equivalent exponential statement. $$\ln .01=-4.6052$$

List the transformations needed to transform the graph of \(h(x)=2^{x}\) into the graph of the given function. $$f(x)=2^{x}-5$$