Chapter 4

A power model has the form _______ .

Match each equation with its model. (a) Exponential growth model (i) \(y=a e^{-b x}, b>0\) (b) Exponential decay model (ii) \(y=a+b \ln x\) (c) Logistic growth model (iii) \(y=\frac{a}{1+b e^{-r x}}\) (d) Gaussian model (iv) \(y=a e^{b x}, b>0\) (e) Natural logarithmic model (v) \(y=a+b \log _{10} x\) (f) Common logarithmic model (vi) \(y=a e^{-(x-b)^{2} / c}\)

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 ___________. (c) \(a^{\log _{a} x}=\) ___________. (b) \(\log _{a} x=\log _{a} y\) if and only if ___________. (d) \(\log _{a} a^{x}=\) ___________.

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

Fill in the blank(s). Exponential and logarithmic functions are examples of nonalgebraic functions, also called _________ functions.

An exponential model of the form \(y=a b^{x}\) can be rewritten as a natural exponential model of the form _________.

A(n) _______ solution does not satisfy the original equation.

Fill in the blank. Gaussian models are commonly used in probability and statistics to represent populations that are _____ distributed.

The base of the ______ logarithmic function is \(10,\) and the base of the ______ logarithmic function is \(e\) .

Fill in the blank(s).Two properties of logarithms are ______ = \(\log _{a} u\) and \(\ln (u v)\) = ______.

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

What type of visual display can you create to get an idea of which type of model will best fit the data set?

What is the value of \(\ln e^{7} ?\)

Fill in the blank. Logistic growth curves are also called ______ curves.

The inverse properties of logarithms are \(\log _{a} a^{x}=x\) and ______ .

Is \(\log _{3} 24=\frac{\ln 3}{\ln 24}\) or \(\log _{3} 24=\frac{\ln 24}{\ln 3}\) correct?

What type of transformation of the graph of \(f(x)=5^{x}\) is the graph of \(f(x+1) ?\)

A power model for a set of data has a coefficient of determination of \(r^{2} \approx 0.901\) and an exponential model for the data has a coefficient of determination of \(r^{2} \approx 0.967 .\) Which model fits the data better?

Can you solve \(5^{x}=125\) using a One-to-One Property?

Which property of logarithms can you use to condense the expression \(\ln x-\ln 2 ?\)

If \(x=e^{y},\) then \(y=\) ______.

The formula \(A=P e^{\pi}\) gives the balance \(A\) of an account earning what type of interest?

Does the model \(y=120 e^{-0.25 x}\) represent exponential growth or exponential decay?

What is the first step in solving the equation \(3+\ln x=10 ?\)

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

What exponential equation is equivalent to the logarithmic equation \(\log _{a} b=c ?\)

Use a calculator to evaluate the function at the indicated value of \(x .\) Round your result to three decimal places. Value \(x=6.8\) \(x=\frac{1}{3}\) \(x=-\pi\) \(x=-\sqrt{2}\) Function \(f(x)=3.4^{x}\)

Do you solve \(\log _{4} x=2\) by using a One-to-One Property or an Inverse Property?

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

For what value(s) of \(x\) is \(\ln x=\ln 7 ?\)

Use a calculator to evaluate the function at the indicated value of \(x .\) Round your result to three decimal places. Value \(x=6.8\) \(x=\frac{1}{3}\) \(x=-\pi\) \(x=-\sqrt{2}\) Function \(f(x)=1.2^{x}\)

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

Rewrite the logarithm as a ratio of (a) common logarithms and (b) natural logarithms.$$\log _{1 / 6} x$$.

Write the logarithmic equation in exponential form. For example, the exponential form of \(\log _{5} 25=2\) is \(5^{2}=25.\) $$\log _{4} 64=3$$

Use a calculator to evaluate the function at the indicated value of \(x .\) Round your result to three decimal places. Value \(x=6.8\) \(x=\frac{1}{3}\) \(x=-\pi\) \(x=-\sqrt{2}\) Function \(g(x)=5^{x}\)

Determine whether each \(x\)-value is a solution of the equation. \(2^{3 x+1}=128\) (a) \(x=-1\) (b) \(x=2\)

Rewrite the logarithm as a ratio of (a) common logarithms and (b) natural logarithms.$$\log _{1 / 4} x$$.

Write the logarithmic equation in exponential form. For example, the exponential form of \(\log _{5} 25=2\) is \(5^{2}=25.\) $$\log _{3} 81=4$$

Use a calculator to evaluate the function at the indicated value of \(x .\) Round your result to three decimal places. Value \(x=6.8\) \(x=\frac{1}{3}\) \(x=-\pi\) \(x=-\sqrt{2}\) Function \(h(x)=8.6^{-3 x}\)

Determine whether each \(x\)-value is a solution of the equation. \(3 e^{x+2}=75\) (a) \(x=-2+e^{25}\) (b) \(x=-2+\ln 25\) (c) \(x \approx 1.2189\)

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

Write the logarithmic equation in exponential form. For example, the exponential form of \(\log _{5} 25=2\) is \(5^{2}=25.\) $$\log _{7} \frac{1}{49}=-2$$

Graph the exponential function by hand. Identify any asymptotes and intercepts and determine whether the graph of the function is increasing or decreasing. $$g(x)=5^{x}$$

Determine whether each \(x\)-value is a 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.$$\log _{a} \frac{4}{5}$$.

Write the logarithmic equation in exponential form. For example, the exponential form of \(\log _{5} 25=2\) is \(5^{2}=25.\) $$\log _{10} \frac{1}{1000}=-3$$

Graph the exponential function by hand. Identify any asymptotes and intercepts and determine whether the graph of the function is increasing or decreasing. $$f(x)=10^{x}$$

Determine whether each \(x\)-value is a solution of the equation. \(\log _{4}(3 x)=3\) (a) \(x \approx 21.3560\) (b) \(x=-4\) (c) \(x=\frac{64}{3}\)

Rewrite the logarithm as a ratio of (a) common logarithms and (b) natural logarithms.$$\log _{2.6} x$$.

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