Chapter 11

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

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

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. \(f(x)=\log _{2} x\) and \(g(x)=\log x\) are examples of ____ functions.

Fill in the blanks. A function is called a _______ function if different inputs determine different outputs.

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. Exponential functions have a constant base and a variable _________________

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. Base-10 logarithms are called ____ logarithms.

Fill in the blanks. The ______ line test can be used to determine whether the graph of a function represents a one-to-one function.

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 \(\square\) = \(\square\) b. If the logarithms base-b of two numbers are equal, the numbers are _____. \(\log _{b} x=\log _{b} y \quad\) is equivalent to \(\square\) = \(\square\)

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

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.

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

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

Fill in the blanks. Like \(\pi,\) the number \(e\) is an__ number. Its decimal representation is non terminating 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. \(\log _{x} 81=4\) is ____ to \(x^{4}=81\).

Fill in the blanks. 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. If \(6^{4 x}=6^{-2},\) then \(4 x= \square\)

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

Refer to the graph shown at the right. a. What type of function is $$ f(x)=3^{x} ? $$ b. What is the domain of the function? c. What is the range of the function? d. What is the \(y\) -intercept of the graph? What is the \(x\) -intercept of the graph? e. Is the function one-to-one? f. What is an asymptote of the graph? g. Is \(f\) an increasing or a decreasing function? h. The graph passes through the point \((1, y) .\) What is \(y ?\) (IMAGE CANNOT COPY)

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

Fill in the blanks. If any horizontal line that intersects the graph of a function does so more than once, the function is not _______

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

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

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. \(T(x)=(0.92)^{x+1}\) f. \(r(x)=x^{3}\) g. \(P(x)=\sqrt{x}\) h. \(d(x)=|x|\)

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 \(f(x)=\) In \(x .\) 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? (GRAPH CANNOT COPY)

Fill in the blanks. 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 \(y\).

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. a. \(3^{-2} \quad\) b. \(\left(\frac{1}{2}\right)^{4} \quad\) c. \(\left(\frac{1}{5}\right)^{-2}\)

Use a calculator to verify that 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. If \(f\) is a one-to-one function, the domain of \(f\) is the _______ \(f^{-1},\) and the range of \(f\) is the _______ of \(f^{-1}\).

Fill in the blanks. a. For \(5^{x}=2,\) the power rule for logarithms provides a way of moving the variable \(x\) from its position as an ______ to a position as a factor. b. If the power rule for logarithms is used on the left side of the equation \(\log 5^{x}=2,\) the resulting equation is \(\square\) \(\log 5=2\)

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?

Fill in the blanks. To two decimal places, the value of \(e\) is ___.

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

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

Fill in the blanks. 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 \(e^{x+2}=4,\) then \(\ln e^{x+2}= \square\)

Fill in the blanks. 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 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. If \(f\) is a one-to-one function, and if \(f(1)=6,\) then \(f^{-1}(6)=\) ____

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

If \(f(x)=x^{2}+3\) and \(g(x)=x-4,\) then \((f / g)(x)=\frac{x^{2}+3}{x-4}\) a. What value of \(x\) makes \(g(x)=0 ?\) b. Fill in the blank: The domain of \(f / g\) is \((-\infty, 4)\) ____ \((4, \infty)\)

Fill in the blanks. $$ \log x=-2 \text { is equivalent to }= $$