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.

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

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

1\. 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?

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

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 \(b>0, b \neq 1\).

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

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

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

When does an extraneous solution occur? How can an extraneous solution be recognized?

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

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?

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. $$ \log _{b}(7 x \cdot 2 y) $$

When can the one-to-one property of logarithms be used to solve an equation? When can it not be used?

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

The graph of \(f(x)=3^{x}\) is refl cted 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.

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

The Oxford Dictionary defi es the word nominal as a value that is "stated or expressed but not necessarily corresponding exactly to the real value. \(^{\mathrm{m}[18]}\) 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.

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."\(^{[18]}\) 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?

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

Defi e Newton's Law of Cooling. Then name at least three real-world situations where Newton's Law of Cooling would be applied.

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

Consider the general logarithmic function \(f(x)=\log _{b}(x) .\) Why can't \(x\) be zero?

Define Newton’s Law of Cooling. Then name at least three real-world situations where Newton’s Law of Cooling would be applied.

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

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 \(\frac{1}{5}.\) What is the equation of the new function,g(x)? State its \(y\) -intercept, domain, and range.

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

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

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

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

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

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

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

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

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. $$f(x)=\log _{3}(x+4)$$

The temperature of an object in degrees Fahrenheit after \(t\) minutes is represented by the equation \(T(t)=68 e^{-0.0174 t}+72 .\) 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. $$ 3^{2 x+1} \cdot 3^{x}=243 $$

The graph of \(f(x)=(1.68)^{x}\) is shifted right 3 units, stretched vertically by a factor of \(2,\) refl cted 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, rewrite each equation in exponential form. $$\log _{4}(q)=m$$

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