Chapter 3

An exponential growth model has the form ________ and an exponential decay model has the form ________.

Fill in the blanks. To _____ an equation in \(x\) means to find all values of \(x\) for which the equation is true.

Fill in the blanks. To evaluate a logarithm to any base, you can use the ________ formula.

The inverse function of the exponential function given by \(f(x)=a^{x}\) is called the _____ function with base \(a\).

Polynomial and rational functions are examples of ________ functions.

A logarithmic model has the form ________ or ________.

Fill in the blanks. To solve exponential and logarithmic equations, you can use the following One- to-One and Inverse Properties. (a) \(a^{x}=a^{y}\) if and only if _____. (b) \(\log _{a} x=\log _{a} y\) if and only if _____. (c) \(a^{\log _{a} x}=\)_____. (d) \(\log _{a} a^{x}=\)_____.

The common logarithmic function has base _____.

Exponential and logarithmic functions are examples of nonalgebraic functions, also called ________ functions.

Gaussian models are commonly used in probability and statistics to represent populations that are ________ ________.

Fill in the blanks. To solve exponential and logarithmic equations, you can use the following strategies. (a) Rewrite the original equation in a form that allows the use of the _____ or logarithmic functions. (b) Rewrite an exponential equation in _____ form and apply the Inverse Property of _____ functions. (c) Rewrite a logarithmic equation in _____ form and apply the Inverse Property of _____ functions.

Fill in the blanks. You can consider \(\log _{a} x\) to be a constant multiple of \(\log _{b} x ;\) the constant multiplier is _____.

The logarithmic function given by \(f(x)=\ln x\) is called the _____ logarithmic function and has base _____.

The graph of a Gaussian model is ________ shaped, where the ________ ________ is the maximum -value of the graph.

The Inverse Properties of logarithms and exponentials state that \(\log _{a} a^{x}=x\) and _____.

The exponential function given by \(f(x)=e^{x}\) is called the ________ ________ function, and the base is called the ________ base.

A logistic growth model has the form ________.

Determine whether each \(x\) -value is a solution (or an approximate solution) of the equation. \(4^{2 x-7}=64\) (a) \(x=5\) (b) \(x=2\)

Match the property of logarithms with its name. (a) Power Property (b) Quotient Property (c) Product Property $$\ln u^{n}=n \ln u$$

The Inverse Properties of logarithms and exponentials state that \(\log _{a} a^{x}=x\) and _____.

A logistic curve is also called a ________ curve.

Determine whether each \(x\) -value is a solution (or an approximate solution) of the equation. \(2^{3 x+1}=32\) (a) \(x=-1\) (b) \(x=2\)

The domain of the natural logarithmic function is the set of _____ _____ _____.

To find the amount \(A\) in an account after \(t\) years with principal \(P\) and an annual interest rate \(r\) compounded continuously, you can use the formula ________.

Determine whether each \(x\) -value is a solution (or an approximate solution) of the equation. \(3 e^{x+2}=75\) (a) \(x=-2+e^{25}\) (b) \(x=-2+\ln 25\) (c) \(x \approx 1.219\)

Rewrite the logarithm as a ratio of (a) common logarithms and (b) natural logarithms. $$\log _{5} 16$$

Write the logarithmic equation in exponential form. For example, the exponential form of \(\log _{5} 25=2\) is \(5^{2}=25\). $$\log _{4} 16=2$$

Evaluate the function at the indicated value of \(x\). Round your result to three decimal places. $$ \begin{aligned} &\text { Function } \quad \text { Valu }\\\ &f(x)=0.9^{x} \quad x=1.4 \end{aligned} $$

Determine whether each \(x\) -value is a solution (or an approximate solution) of the equation. \(4 e^{x-1}=60\) (a) \(x=1+\ln 15\) (b) \(x \approx 3.7081\) (c) \(x=\ln 16\)

Rewrite the logarithm as a ratio of (a) common logarithms and (b) natural logarithms. $$\log _{3} 47$$

Write the logarithmic equation in exponential form. For example, the exponential form of \(\log _{5} 25=2\) is \(5^{2}=25\). $$\log _{7} 343=3$$

Evaluate the function at the indicated value of \(x\). Round your result to three decimal places. $$ \begin{aligned} &\text { Function } \quad \text { Valu }\\\ &f(x)=2.3^{x} \quad x=\frac{3}{2} \end{aligned} $$

Determine whether each \(x\) -value is a solution (or an approximate solution) of the equation. \(\log _{4}(3 x)=3\) (a) \(x \approx 21.333\) (b) \(x=-4\) (c) \(x=\frac{64}{3}\)

Rewrite the logarithm as a ratio of (a) common logarithms and (b) natural logarithms. $$\log _{1 / 5} x$$

Write the logarithmic equation in exponential form. For example, the exponential form of \(\log _{5} 25=2\) is \(5^{2}=25\). $$\log _{9} \frac{1}{81}=-2$$

Evaluate the function at the indicated value of \(x\). Round your result to three decimal places. $$ \begin{aligned} &\text { Function } \quad \text { Valu }\\\ &f(x)=5^{x} \quad x=-\pi \end{aligned} $$

Determine whether each \(x\) -value is a solution (or an approximate solution) of the equation. \(\log _{2}(x+3)=10\) (a) \(x=1021\) (b) \(x=17\) (c) \(x=10^{2}-3\)

Rewrite the logarithm as a ratio of (a) common logarithms and (b) natural logarithms. $$\log _{1 / 3} x$$

Write the logarithmic equation in exponential form. For example, the exponential form of \(\log _{5} 25=2\) is \(5^{2}=25\). $$\log \frac{1}{1000}=-3$$

Evaluate the function at the indicated value of \(x\). Round your result to three decimal places. $$ \begin{aligned} &\text { Function } \quad \text { Valu }\\\ &f(x)=\left(\frac{2}{3}\right)^{5 x} \quad x=\frac{3}{10} \end{aligned} $$

Determine whether each \(x\) -value is a solution (or an approximate solution) of the equation. \(\ln (2 x+3)=5.8\) (a) \(x=\frac{1}{2}(-3+\ln 5.8)\) (b) \(x=\frac{1}{2}\left(-3+e^{5.8}\right)\) (c) \(x \approx 163.650\)

Rewrite the logarithm as a ratio of (a) common logarithms and (b) natural logarithms. $$\log _{x} \frac{3}{10}$$

Write the logarithmic equation in exponential form. For example, the exponential form of \(\log _{5} 25=2\) is \(5^{2}=25\). $$\log _{32} 4=\frac{2}{5}$$

Evaluate the function at the indicated value of \(x\). Round your result to three decimal places. $$ \begin{aligned} &\text { Function } \quad \text { Valu }\\\ &g(x)=5000\left(2^{x}\right) \quad x=-1.5 \end{aligned} $$

Determine whether each \(x\) -value is a solution (or an approximate solution) of the equation. \(\ln (x-1)=3.8\) (a) \(x=1+e^{3.8}\) (b) \(x \approx 45.701\) (c) \(x=1+\ln 3.8\)

Rewrite the logarithm as a ratio of (a) common logarithms and (b) natural logarithms. $$\log _{x} \frac{3}{4}$$

Write the logarithmic equation in exponential form. For example, the exponential form of \(\log _{5} 25=2\) is \(5^{2}=25\). $$\log _{16} 8=\frac{3}{4}$$

Evaluate the function at the indicated value of \(x\). Round your result to three decimal places. $$ \begin{aligned} &\text { Function } \quad \text { Valu }\\\ &f(x)=200(1.2)^{12 x} \quad x=24 \end{aligned} $$

(a) solve for \(P\) and (b) solve for \(t\). $$A=P e^{r t}$$

Solve for \(x\). $$4^{x}=16$$