Chapter 6

For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth. \(\log (\sqrt{2})\)

For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth. \(\ln (\sqrt{2})\)

Recall that an exponential function is any equation written in the form \(f(x)=a \cdot b^{x}\) such that \(a\) and \(b\) are positive numbers and \(b \neq 1\). Any positive number \(b\) can be written as \(b=e^{n}\) for some value of \(n .\) Use this fact to rewrite the formula for an exponential function that uses the number \(e\) as a base.

Find the inverse function \(f^{-1}(x)\) for the logistic function \(f(x)=\frac{c}{1+a e^{-b x}} .\) Show all steps.

For the following exercises, solve the equation for \(x\), if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. \(\log _{11}\left(-2 x^{2}-7 x\right)=\log _{11}(x-2)\)

What is the domain of the function \(f(x)=\ln \left(\frac{x+2}{x-4}\right) ?\) Discuss the result.

In an exponential decay function, the base of the exponent is a value between 0 and \(1 .\) Thus, for some number \(b>1,\) the exponential decay function can be written as \(f(x)=a \cdot\left(\frac{1}{b}\right)^{x} .\) Use this formula, along with the fact that \(b=e^{n},\) to show that an exponential decay function takes the form \(f(x)=a(e)^{-n x}\) for some positive number \(n\).

Use properties of exponents to find the \(x\) -intercepts of the function \(f(x)=\log \left(x^{2}+4 x+4\right)\) algebraically. Show the steps for solving, and then verify the result by graphing the function.

Is \(f(x)=0\) in the range of the function \(f(x)=\log (x) ?\) If so, for what value of \(x ?\) Verify the result.

The formula for the amount \(A\) in an investment account with a nominal interest rate \(r\) at any time \(t\) is given by \(A(t)=a(e)^{r t},\) where \(a\) is the amount of principal initially deposited into an account that compounds continuously. Prove that the percentage of interest earned to principal at any time \(t\) can be calculated with the formula \(I(t)=e^{r t}-1\)

For the following exercises, solve the equation for \(x\), if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. \(\log _{9}(3-x)=\log _{9}(4 x-8)\)

Is there a number \(x\) such that \(\ln x=2 ?\) If so, what is that number? Verify the result.

The fox population in a certain region has an annual growth rate of \(9 \%\) per year. In the year 2012, there were 23,900 fox counted in the area. What is the fox population predicted to be in the year \(2020 ?\)

For the following exercises, solve the equation for \(x\), if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. \(\log \left(x^{2}+13\right)=\log (7 x+3)\)

Is the following true: \(\frac{\log _{3}(27)}{\log _{4}\left(\frac{1}{64}\right)}=-1 ?\) Verify the result.

A scientist begins with 100 milligrams of a radioactive substance that decays exponentially. After 35 hours, \(50 \mathrm{mg}\) of the substance remains. How many milligrams will remain after 54 hours?

For the following exercises, solve the equation for \(x\), if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. \(\frac{3}{\log _{2}(10)}-\log (x-9)=\log (44)\)

In the year \(1985,\) a house was valued at \(\$ 110,000\). By the year 2005, the value had appreciated to \(\$ 145,000 .\) What was the annual growth rate between 1985 and \(2005 ?\) Assume that the value continued to grow by the same percentage. What was the value of the house in the year \(2010 ?\)

For the following exercises, solve the equation for \(x\), if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. \(\ln (x)-\ln (x+3)=\ln (6)\)

A car was valued at \(\$ 38,000\) in the year 2007. By 2013, the value had depreciated to \(\$ 11,000\) If the car's value continues to drop by the same percentage, what will it be worth by \(2017 ?\)

For the following exercises, solve for the indicated value, and graph the situation showing the solution point. An account with an initial deposit of \(\quad \$ 6,500\) earns 7.25\% annual interest, compounded continuously. How much will the account be worth after 20 years?

Jamal wants to save \(\$ 54,000\) for a down payment on a home. How much will he need to invest in an account with \(8.2 \% \mathrm{APR}\) compounding daily, in order to reach his goal in 5 years?

For the following exercises, solve for the indicated value, and graph the situation showing the solution point. The formula for measuring sound intensity in decibels \(D\) is defined by the equation \(D=10 \log \left(\frac{I}{I_{0}}\right),\) where \(I\) is the intensity of the sound in watts per square meter and \(I_{0}=10^{-12}\) is the lowest level of sound that the average person can hear. How many decibels are emitted from a jet plane with a sound intensity of \(8.3 \cdot 10^{2}\) watts per square meter?

The intensity levels \(I\) of two earthquakes measured on a seismograph can be compared by the formula \(\log \frac{I_{1}}{I_{2}}=M_{1}-M_{2}\) where \(M\) is the magnitude given by the Richter Scale. In August \(2009, \mathrm{an}\) earthquake of magnitude 6.1 hit Honshu, Japan. In March 2011, that same region experienced yet another, more devastating earthquake, this time with a magnitude of \(9.0 .{ }^{8}\) How many times greater was the intensity of the 2011 earthquake? Round to the nearest whole number.

Kyoko has \(\$ 10,000\) that she wants to invest. Her bank has several investment accounts to choose from, all compounding daily. Her goal is to have \(\$ 15,000\) by the time she finishes graduate school in 6 years. To the nearest hundredth of a percent, what should her minimum annual interest rate be in order to reach her goal? (Hint solve the compound interest formula for the interest rate.)

For the following exercises, solve for the indicated value, and graph the situation showing the solution point. The population of a small town is modeled by the equation \(P=1650 e^{0.5 t}\) where \(t\) is measured in years. In approximately how many years will the town's population reach \(20,000 ?\)

Alyssa opened a retirement account with \(7.25 \%\) APR in the year 2000. Her initial deposit was \(\$ 13,500\). How much will the account be worth in 2025 if interest compounds monthly? How much more would she make if interest compounded continuously?

For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate the variable to 3 decimal places. \(1000(1.03)^{t}=5000\) using the common log.

An investment account with an annual interest rate of \(7 \%\) was opened with an initial deposit of $$\$ 4,000$$ Compare the values of the account after 9 years when the interest is compounded annually, quarterly, monthly, and continuously.

For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate the variable to 3 decimal places. \(e^{5 x}=17\) using the natural log

For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate the variable to 3 decimal places. \(3(1.04)^{3 t}=8\) using the common log

For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate the variable to 3 decimal places. \(3^{4 x-5}=38\) using the common log

For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate the variable to 3 decimal places. \(50 e^{-0.12 t}=10\) using the natural log

For the following exercises, use a calculator to solve the equation. Unless indicated otherwise, round all answers to the nearest ten-thousandth. \(7 e^{3 x-5}+7.9=47\)

For the following exercises, use a calculator to solve the equation. Unless indicated otherwise, round all answers to the nearest ten-thousandth. \(\ln (3)+\ln (4.4 x+6.8)=2\)

Atmospheric pressure \(P\) in pounds per square inch is represented by the formula \(P=14.7 e^{-0.21 x},\) where \(x\) is the number of miles above sea level. To the nearest foot, how high is the peak of a mountain with an atmospheric pressure of 8.369 pounds per square inch? (Hint: there are 5280 feet in a mile)

The magnitude \(M\) of an earthquake is represented by the equation \(M=\frac{2}{3} \log \left(\frac{E}{E_{0}}\right)\) where \(E\) is the amount of energy released by the earthquake in joules and \(E_{0}=10^{4.4}\) is the assigned minimal measure released by an earthquake. To the nearest hundredth, what would the magnitude be of an earthquake releasing \(1.4 \cdot 10^{13}\) joules of energy?

Use the definition of a logarithm along with the oneto-one property of logarithms to prove that \(b^{\log _{b} x}=x\).

Recall the formula for continually compounding interest, \(y=A e^{k t} .\) Use the definition of a logarithm along with properties of logarithms to solve the formula for time \(t\) such that \(t\) is equal to a single logarithm.

Recall the compound interest formula \(A=a\left(1+\frac{r}{k}\right)^{k t} .\) Use the definition of a logarithm along with properties of logarithms to solve the formula for time \(t\)

Newton's Law of Cooling states that the temperature \(T\) of an object at any time \(t\) can be described by the equation \(T=T_{s}+\left(T_{0}-T_{s}\right) e^{-k t},\) where \(T_{s}\) is the temperature of the surrounding environment, \(T_{0}\) is the initial temperature of the object, and \(k\) is the cooling rate. Use the definition of a logarithm along with properties of logarithms to solve the formula for time \(t\) such that \(t\) is equal to a single logarithm.