Chapter 3

Expanding a Logarithmic Expression In Exercises \(37-58\) , use the properties of logarithms to expand the expression as a sum, difference, and or constant multiple of logarithms. (Assume all variables are positive.) $$\ln \sqrt[4]{x^{3}\left(x^{2}+3\right)}$$

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

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

Polynomial and rational functions are examples of______________functions.

An exponential growth model has the form ________, and an exponential decay model has the form _________

Fill in the blanks. An _____ solution does not satisfy the original equation.

Fill in the blanks. The common logarithmic function has base _______ .

Exponential and logarithmic functions are examples of nonalgebraic functions, also called____________functions.

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

In Exercises \(1-3,\) fill in the blanks. You can consider log \(_{a} x\) to be a constant multiple of \(\log _{b} x ;\) the constant multiplier is _____ .

Fill in the blanks. The logarithmic function \(f(x)=\ln x\) is called the ________ logarithmic function and has base ________ .

You can use the___________ Property to solve simple exponential equations.

In probability and statistics, Gaussian models commonly represent populations that are ________ ________.

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) \(x=\ln 16\)

Fill in the blanks. The Inverse Properties of logarithms state that \(\log _{a} a^{x}=x\) and _________ .

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

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) \(x=10^{2}-3\)

Fill in the blanks. The One-to-One Property of natural logarithms states that if \(\ln x=\ln y,\) then _________ .

To find the amount \(A\) in an account after \(t\) years with principal \(P\) and an annual interest rate \(r\) compounded \(n\) times per year, you can use the formula________________.

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) \(x \approx 163.650\)

Fill in the blanks. 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______________.

A=P\left(1+\frac{r}{n}\right)^{n t}

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

Rewriting a Logarithm In Exercises \(7-10\) , 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 \(5^{2}=25 .\) \(\log _{4} 16=2\)

Evaluating an Exponential Function In Exercises \(7-12\) , evaluate the function at the indicated value of \(x .\) Round your result to three decimal places. Function Value \(f(x)=0.9^{x}\) \(x=1.4\)

Rewriting a Logarithm In Exercises \(7-10\) , rewrite the logarithm as a ratio of (a) common logarithms and (b) natural logarithms. $$\log _{1 / 5} x$$

Evaluating an Exponential Function In Exercises \(7-12\) , evaluate the function at the indicated value of \(x .\) Round your result to three decimal places. $$Function$$ $$f(x)=0.9^{x}$$ $$Value$$ $$x=1.4$$

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

Rewriting a Logarithm In Exercises \(7-10\) , rewrite the logarithm as a ratio of (a) common logarithms and (b) natural logarithms. $$\log _{x} \frac{3}{10}$$

Solve for \(x.\) \(\ln x-\ln 2=0\)

Evaluating an Exponential Function In Exercises \(7-12,\) evaluate the function at the indicated value of \(x .\) Round your result to three decimal places. $$Function$$ $$f(x)=5^{x}$$ $$Value$$ $$x=-\pi$$

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

Rewriting a Logarithm In Exercises \(7-10\) , rewrite the logarithm as a ratio of (a) common logarithms and (b) natural logarithms. $$\log _{2.6} x$$

Solve for \(x.\) \(e^{x}=2\)

Evaluating an Exponential Function In Exercises \(7-12,\) evaluate the function at the indicated value of \(x .\) Round your result to three decimal places. $$Function$$ $$f(x)=\left(\frac{2}{3}\right)^{5 x}$$ $$Value$$ $$x=\frac{3}{10}$$

Write the logarithmic equation in exponential form. For example, the exponential form of \(\log _{5} 25=2\) is \(5^{2}=25 .\) \(\log _{64} 8=\frac{1}{2}\)

Using the Change-of-Base Formula In Exercises \(11-14\) , evaluate the logarithm using the change-of-base formula. Round your result to three decimal places. $$\log _{4} 8$$

Solve for \(x.\) \(\ln x=-1\)

Evaluating an Exponential Function In Exercises \(7-12,\) evaluate the function at the indicated value of \(x .\) Round your result to three decimal places. $$Function$$ $$g(x)=5000\left(2^{x}\right)$$ $$Value$$ $$x=-1.5$$

Write the exponential equation in logarithmic form. For example, the logarithmic form of \(2^{3}=8\) is log \(_{2} 8=3\) \(5^{3}=125\)

Using the Change-of-Base Formula In Exercises \(11-14\) , evaluate the logarithm using the change-of-base formula. Round your result to three decimal places. $$\log _{1 / 2} 4$$

Solve for \(x.\) \(\log x=-2\)

Evaluating an Exponential Function In Exercises \(7-12,\) evaluate the function at the indicated value of \(x .\) Round your result to three decimal places. $$Function$$ $$f(x)=200(1.2)^{12 x}$$ $$Value$$ $$x=24$$

Write the exponential equation in logarithmic form. For example, the logarithmic form of \(2^{3}=8\) is log \(_{2} 8=3\) \(9^{3 / 2}=27\)

Using the Change-of-Base Formula In Exercises \(11-14,\) evaluate the logarithm using the change-of-base formula. Round your result to three decimal places. $$\log _{9} 0.1$$

Solve for \(x.\) \(\log _{4} x=3\)

Write the exponential equation in logarithmic form. For example, the logarithmic form of \(2^{3}=8\) is log \(_{2} 8=3\) \(4^{-3}=\frac{1}{64}\)

Using the Change-of-Base Formula In Exercises \(11-14,\) evaluate the logarithm using the change-of-base formula. Round your result to three decimal places. $$\log _{3} 0.015$$