Chapter 9

Fill in the blanks. The _____ of \(f\) and \(g,\) denoted as \(f+g,\) is defined by \((f+g)(x)=\square\) and the _____ of \(f\) and \(g\), denoted as \(f-g,\) is defined by \((f-g)(x)=\square\).

Fill in the blanks. \(f(x)=e^{x}\) is called the natural ______ function. The base is _____

Fill in the blanks. The logarithm of a _____ such as \(\log _{3} 4 x,\) equals the sum of the logarithms of the factors.

Fill in the blanks. An equation with a positive constant base and a variable in its exponent, such as \(3^{2 x}=8,\) is called an ___ equation.

Fill in the blanks. \(f(x)=\log _{2} x\) and \(g(x)=\log x\) are examples of _____ functions.

Fill in the blanks. \(f(x)=2^{x}\) and \(f(x)=\left(\frac{1}{4}\right)^{x}\) are examples of _______ functions.

A function is called a _________ function if different inputs determine different outputs.

Fill in the blanks. \(f(x)=\ln x\) is called the ______ logarithmic function. The base is _______.

Fill in the blanks. The logarithm of a _____ such as \(\log _{2} \frac{5}{x},\) equals the difference of the logarithms of the numerator and denominator.

Fill in the blanks. An equation with a logarithmic expression that contains a variable, such as \(\log _{5}(2 x-3)=\log _{5}(x+4),\) is a ___equation.

Fill in the blanks. Base-10 logarithms are called _____ logarithms.

Exponential functions have a constant base and a variable __________

The _________ line test can be used to determine whether the graph of a function represents a one-to-one function.

Fill in the blanks. The _____ of the function \(f+g\) is the set of real numbers \(x\) that are in the domain of both \(f\) and \(g .\)

If a bank pays interest infinitely many times a year, we say that the interest is compounded _______

Fill in the blanks. a. If two exponential expressions with the same base are equal, their exponents are ___. \(b^{x}=b^{y} \quad\) is equivalent to __=__. b. If the logarithms base-b of two numbers are equal, the numbers are ___. $$ \log _{b} x=\log _{b} y $$ is equivalent to __=__.

Fill in the blanks. The graph of \(f(x)=\log _{2} x\) approaches, but never touches, the negative portion of the \(y\) -axis. Thus the \(y\) - axis is an _____ of the graph.

The graph of \(f(x)=3^{x}\) approaches, but never touches, the negative portion of the \(x\) -axis. Thus, the \(x\) -axis is an __________ of the graph.

Fill in the blanks. Like \(\pi,\) the number \(e\) is an ____ number. Its decimal representation is nonterminating and ______

Fill in the blanks. The _____ - of-base formula converts a logarithm of one base to a ratio of logarithms of a different base.

Fill in the blanks. The right side of the exponential equation \(5^{x-3}=125\) can be written as a power of ___.

Fill in the blanks. \(\log _{x} 81=4\) is _____ to \(x^{4}=81.\)

______ interest is paid on the principal and previously earned interest.

The graphs of a function and its inverse are _________ images of each other with respect to \(y=x .\) We also say that their graphs are ___________ with respect to the line \(y=x\)

Fill in the blanks. When we write \((f \circ g)(x)\) as \(f(g(x)),\) we have changed from \(\circ\) notation to _____ parentheses notation.

Fill in the blanks. In problem \(6,\) also give the name of each rule. a. \(\log _{b} 1=\square\) b. \(\log _{b} b=\square\) c. \(\log _{b} b^{x}=\square\) \(b^{\log _{b} x}=\square\)

Fill in the blanks. $$ \text { If } 6^{4 x}=6^{-2}, \text { then } 4 x= $$

Refer to the graph on the right. GRAPH CANNOT COPY a. What type of function is\(f(x)=\log _{4} x ?\) b. What is the domain of the function? What is the range of the function? c. What is the \(y\) -intercept of the graph? What is the \(x\) -intercept of the graph? d. Is \(f\) a one-to-one function? e. What is an asymptote of the graph? I. Is \(f\) an increasing or a decreasing function? g. The graph passes through the point \((4, y) .\) What is \(y ?\)

If any horizontal line that intersects the graph of a function does so more than once, the function is not _________

Fill in the blanks. When reading the notation \(f(g(x)),\) we say "f _____ g _____ x ".

a. Use a calculator to complete the table of values in the next column for \(f(x)=\ln x .\) Round to the nearest hundredth. b. Graph \(f(x)=\ln x .\) Note that the units on the \(x\) - and \(y\) -axes are different. c. What are the domain and range of the function? d. What is the \(x\) -intercept of the graph? What is the \(y\) -intercept? e. What is an asymptote of the graph? f. Is \(f\) increasing or decreasing? g. Is the function one-to-one? $$ \begin{array}{|c|c|} \hline \boldsymbol{x} & \boldsymbol{f}(\boldsymbol{x}) \\ \hline 0.5 & \\ 1 & \\ 2 & \\ 4 & \\ 6 & \\ 8 & \\ 10 & \\ \hline \end{array} $$ (Graph can't copy)

Fill in the blanks. In problem \(6,\) also give the name of each rule. a. \(\log _{b} M N=\log _{b}+\log _{b}\square\)_____ rule b. \(\log _{b} \frac{M}{N}=\log _{b} M\square\) \(\log _{b} N\)_____ rule c. \(\log _{b} M^{P}=p \log _{b}\square\) _____ rule d. \(\log _{b} x=\frac{\log _{a}\square}{\log\square b}\)

Fill in the blanks. a. Write the equivalent base-10 exponential equation for \(\log (x+1)=2\) b. Write the equivalent base- \(e\) exponential equation for \(\ln (x+1)=2\)

Which of the following functions are exponential functions? a. \(f(x)=x^{2}\) b. \(g(x)=4 x\) c. \(h(x)=8^{x}\) d. \(s(x)=\frac{1}{x}\) e. \(T(x)=(0.92)^{x+1}\) t. \(r(x)=x^{3}\) g. \(P(x)=\sqrt{x}\) h. \(d(x)=|x|\)

To find the inverse of the function \(f(x)=2 x-3,\) we begin by replacing \(f(x)\) with \(y,\) and then we _________ x and \(y\)

Use a calculator to verify that each equation is true. See Using Your Calculator: Verifying Properties of Logarithms. $$ \log (2.5 \cdot 3.7)=\log 2.5+\log 3.7 $$

Fill in the blanks. To solve \(5^{x}=2\), we can take the ___ of both sides of the equation to get \(\log 5^{x}=\log 2\)

Evaluate each expression without a calculator. $$ \begin{aligned} &\text { a. } 3^{-2} \quad \text { b. }\left(\frac{1}{2}\right)^{4}\\\ &\text { c. }\left(\frac{1}{5}\right)^{-2} \end{aligned} $$

If \(f\) is a one-to-one function, the domain of \(f\) is the _________ of \(f^{-1},\) and the range of \(f\) is the _______ of \(f^{-1}\)

a. If \(f(x)=3 x+1\) and \(g(x)=1-2 x,\) find \(f(g(3))\) and \(g(f(3))\) b. Is the composition of functions commutative?

To two decimal places, the value of \(e\) is _____.

Use a calculator to verify that each equation is true. See Using Your Calculator: Verifying Properties of Logarithms. $$ \ln (2.25)^{4}=4 \ln 2.25 $$

Evaluate each expression using a calculator. Round to the nearest tenth. a. \(20,000(1.036)^{52}\) b. \(92(0.88)^{6}\)

If a function turns an input of 2 into an output of \(5,\) the inverse function will turn an input of 5 into the output

If \(n\) gets larger and larger, the value of \(\left(1+\frac{1}{n}\right)^{n}\) approaches the value of ________.

Use a calculator to verify that each equation is true. See Using Your Calculator: Verifying Properties of Logarithms. $$ \ln \frac{11.3}{6.1}=\ln 11.3-\ln 6.1 $$

Fill in the blanks. $$ \text { If } e^{x+2}=4, \text { then } \ln e^{x+2}= $$

Fill in the blanks. \(\log _{6} 36=2\) means____ \(=\)_____

If \(f\) is a one-to-one function, and if \(f(1)=6,\) then \(f^{-1}(6)=\)

Fill in the blanks. Perform a check to determine whether \(-2\) is a solution of \(5^{2 x+3}=\frac{1}{5}\)