Chapter 6

What situations are best modeled by a logistic equation? Give an example, and state a case for why the example is a good fit.

With what kind of exponential model would half-life be associated? What role does halflife play in these models?

How does the power rule for logarithms help when solving logarithms with the form \(\log _{b}(\sqrt[n]{x}) ?\)

The inverse of every logarithmic function is an exponential function and vice- versa. What does this tell us about the relationship between the coordinates of the points on the graphs of each?

What is a base \(b\) logarithm? Discuss the meaning by interpreting each part of the equivalent equations \(b^{y}=x\) and \(\log _{b} x=y\) for \(b>0, b \neq 1\).

What role does the horizontal asymptote of an exponential function play in telling us about the end behavior of the graph?

Explain why the values of an increasing exponential function will eventually overtake the values of an increasing linear function.

What is a carrying capacity? What kind of model has a carrying capacity built into its formula? Why does this make sense?

What is carbon dating? Why does it work? Give an example in which carbon dating would be useful.

What does the change-of-base formula do? Why is it useful when using a calculator?

What type(s) of translation(s), if any, affect the range of a logarithmic function?

How is the logarithmic function \(f(x)=\log _{b} x\) related to the exponential function \(g(x)=b^{x} ?\) What is the result of composing these two functions?

What is the advantage of knowing how to recognize transformations of the graph of a parent function algebraically?

Given a formula for an exponential function, is it possible to determine whether the function grows or decays exponentially just by looking at the formula? Explain.

For the following exercises, expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs. \(\log _{b}(7 x \cdot 2 y)\)

How can the logarithmic equation \(\log _{b} x=y\) be solved for \(x\) using the properties of exponents?

The graph of \(f(x)=3^{x}\) is reflected about the \(y\) -axis and stretched vertically by a factor of 4\. What is the equation of the new function, \(g(x) ?\) State its \(y\) -intercept, domain, and range.

The Oxford Dictionary defines the word nominal as a value that is "stated or expressed but not necessarily corresponding exactly to the real value. \(^{\prime \prime}\) Develop a reasonable argument for why the term nominal rate is used to describe the annual percentage rate of an investment account that compounds interest.

What might a scatterplot of data points look like if it were best described by a logarithmic model?

Define Newton's Law of Cooling. Then name at least three realworld situations where Newton's Law of Cooling would be applied.

For the following exercises, use like bases to solve the exponential equation. \(4^{-3 v-2}=4^{-v}\)

For the following exercises, expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs. \(\ln (3 a b \cdot 5 c)\)

Consider the general logarithmic function \(f(x)=\log _{b}(x)\). Why \(\operatorname{can}^{\prime} \mathrm{t} x\) be zero?

Discuss the meaning of the common logarithm. What is its relationship to a logarithm with base \(b\), and how does the notation differ?

The graph of \(f(x)=\left(\frac{1}{2}\right)^{-x}\) is reflected about the \(y\) -axis and compressed vertically by a factor of \(\frac{1}{5} . \quad\) What is the equation of the new function, \(g(x) ?\) State its \(y\) -intercept, domain, and range.

For the following exercises, identify whether the statement represents an exponential function. Explain. The average annual population increase of a pack of wolves is \(25 .\)

What does the \(y\) -intercept on the graph of a logistic equation correspond to for a population modeled by that equation?

What is an order of magnitude? Why are orders of magnitude useful? Give an example to explain.

For the following exercises, use like bases to solve the exponential equation. \(64 \cdot 4^{3 x}=16\)

For the following exercises, expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs. \(\log _{b}\left(\frac{13}{17}\right)\)

Does the graph of a general logarithmic function have a horizontal asymptote? Explain.

Discuss the meaning of the natural logarithm. What is its relationship to a logarithm with base \(b\), and how does the notation differ?

The graph of \(f(x)=10^{x}\) is reflected about the \(x\) -axis and \(\begin{array}{l}\text { shifted upward } 7 & \text { units. What }\end{array}\) is the equation of the new function, \(g(x) ?\) State its \(y\) -intercept, domain, and range.

For the following exercises, identify whether the statement represents an exponential function. Explain. A population of bacteria decreases by a factor of \(\frac{1}{8}\) every 24 hours.

The temperature of an object in degrees Fahrenheit after \(t\) minutes is represented by the equation \(T(t)=68 e^{-0.0174 t}+72 . \quad\) To the nearest degree, what is the temperature of the object after one and a half hours?

For the following exercises, use like bases to solve the exponential equation. \(3^{2 x+1} \cdot 3^{x}=243\)

For the following exercises, expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs. \(\log _{4}\left(\frac{\frac{x}{z}}{w}\right)\)

For the following exercises, state the domain and range of the function. \(f(x)=\log _{3}(x+4)\)

For the following exercises, rewrite each equation in exponential form. \(\log _{4}(q)=m\)

The graph of \(f(x)=(1.68)^{x}\) is shifted right 3 units, stretched vertically by a factor of 2, reflected about the \(x\) -axis, and then shifted downward 3 units. What is the equation of the new function, \(g(x) ?\) State its \(y\) -intercept (to the nearest thousandth), domain, and range.

For the following exercises, identify whether the statement represents an exponential function. Explain. The value of a coin collection has increased by \(3.25 \%\) annually over the last 20 years.

For the following exercises, use the logistic growth model \(f(x)=\frac{150}{1+8 e^{-2 x}}\). Find and interpret \(f(0)\). Round to the nearest tenth.

For the following exercises, use like bases to solve the exponential equation. \(2^{-3 n} \cdot \frac{1}{4}=2^{n+2}\)

For the following exercises, expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs. \(\ln \left(\frac{1}{4^{k}}\right)\)

For the following exercises, state the domain and range of the function. \(h(x)=\ln \left(\frac{1}{2}-x\right)\)

For the following exercises, rewrite each equation in exponential form. \(\log _{a}(b)=c\)

The graph of \(f(x)=2\left(\frac{1}{4}\right)^{x-20}\) is shifted downward 4 units, and then shifted left 2 units, stretched vertically by a factor of \(4,\) and reflected about the \(x\) -axis. What is the equation of the new function, \(g(x) ?\) State its \(y\) -intercept, domain, and range.

For the following exercises, identify whether the statement represents an exponential function. Explain. For each training session, a personal trainer charges his clients \(\$ 5\) less than the previous training session.

For the following exercises, use the logistic growth model \(f(x)=\frac{150}{1+8 e^{-2 x}}\). Find and interpret \(f(4)\). Round to the nearest tenth.

For the following exercises, use like bases to solve the exponential equation. \(625 \cdot 5^{3 x+3}=125\)