Chapter 4

Evaluate or simplify each expression without using a calculator. $$ \ln 1 $$

In Exercises \(83-88,\) let \(\log _{b} 2=A\) and \(\log _{b} 3=\) C. Write each expression in terms of \(A\) and \(C\). $$ \log _{b} \sqrt{\frac{2}{27}} $$

Solve each logarithmic equation in Exercises \(49-92 .\) Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ \log x+\log (x+3)=\log 10 $$

In Exercises \(87-90,\) determine whether cach statement is true or false If the statement is false, make the necessary change(s) to produce a true statement. As the number of compounding periods increases on a fixed investment, the amount of money in the account over a fixed interval of time will increase without bound.

Evaluate or simplify each expression without using a calculator. $$ \ln e $$

Solve each logarithmic equation in Exercises \(49-92 .\) Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ \log (x+3)+\log (x-2)=\log 14 $$

Evaluate or simplify each expression without using a calculator. $$ \ln e^{6} $$

In Exercises \(89-102,\) determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. $$ \ln e=0 $$

Solve each logarithmic equation in Exercises \(49-92 .\) Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ \ln (x-4)+\ln (x+1)=\ln (x-8) $$

Evaluate or simplify each expression without using a calculator. $$ \ln e^{7} $$

Solve each logarithmic equation in Exercises \(49-92 .\) Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ \log _{2}(x-1)-\log _{2}(x+3)=\log _{2}\left(\frac{1}{x}\right) $$

Evaluate or simplify each expression without using a calculator. $$ \ln \frac{1}{e^{6}} $$

In Exercises \(89-102,\) determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. $$ \log _{4}\left(2 x^{3}\right)=3 \log _{4}(2 x) $$

Solve each logarithmic equation in Exercises \(49-92 .\) Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ \ln (x-2)-\ln (x+3)=\ln (x-1)-\ln (x+7) $$

Evaluate or simplify each expression without using a calculator. $$ \ln \frac{1}{e^{7}} $$

Solve each logarithmic equation in Exercises \(49-92 .\) Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ \ln (x-5)-\ln (x+4)=\ln (x-1)-\ln (x+2) $$

Graph \(f(x)-2^{x}\) and its inverse function in the same rectangular coordinate system.

Evaluate or simplify each expression without using a calculator. $$ e^{\ln 125} $$

Solve each equation. $$ 5^{2 x} \cdot 5^{4 x}=125 $$

The hyperbolic cosine and hyperbolic sine functions are defined by $$ \cosh x-\frac{e^{x}+e^{-x}}{2} \text { and } \sinh x-\frac{e^{x}-e^{-x}}{2} $$ a. Show that \(\cosh x\) is an even function. b. Show that \(\sinh x\) is an odd function. c. Prove that \((\cosh x)^{2}-(\sinh x)^{2}-1\)

Evaluate or simplify each expression without using a calculator. $$ e^{\ln 300} $$

In Exercises \(89-102,\) determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. $$ \ln (x+1)=\ln x+\ln 1 $$

Solve each equation. $$ 3^{x+2} \cdot 3^{x}=81 $$

Evaluate or simplify each expression without using a calculator. $$ \ln e^{9 x} $$

In Exercises \(89-102,\) determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. $$ \ln (5 x)+\ln 1=\ln (5 x) $$

Solve each equation. $$ 2|\ln x|-6=0 $$

Exercises \(94-96\) will help you prepare for the material covered in the next section. 25 to what power gives \(5 ?\left(25^{7}-5\right)\)

Evaluate or simplify each expression without using a calculator. $$ \ln e^{13 x} $$

Solve each equation. $$ 3|\log x|-6=0 $$

Exercises \(94-96\) will help you prepare for the material covered in the next section. $$ \text { Solve: }(x-3)^{2}>0 $$

Evaluate or simplify each expression without using a calculator. $$ e^{\ln 5 x^{2}} $$

In Exercises \(89-102,\) determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. $$ \log (x+3)-\log (2 x)=\frac{\log (x+3)}{\log (2 x)} $$

Evaluate or simplify each expression without using a calculator. $$ e^{\ln 7 x^{2}} $$

In Exercises \(89-102,\) determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. $$ \log _{6}\left(\frac{x-1}{x^{2}+4}\right)=\log _{6}(x-1)-\log _{6}\left(x^{2}+4\right) $$

Solve each equation. $$ \ln (2 x+1)+\ln (x-3)-2 \ln x=0 $$

In Exercises \(89-102,\) determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. $$ \log _{6}[4(x+1)]=\log _{6} 4+\log _{6}(x+1) $$

Solve each equation. $$ \ln 3-\ln (x+5)-\ln x=0 $$

Write each equation in its equivalent exponential form. Then solve for \(x .\) $$ \log _{3}(x-1)=2 $$

In Exercises \(89-102,\) determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. $$ \log _{3} 7=\frac{1}{\log _{7} 3} $$

Solve each equation. $$ 5^{x^{2}-12}=25^{2 x} $$

Write each equation in its equivalent exponential form. Then solve for \(x .\) $$ \log _{5}(x+4)=2 $$

In Exercises \(89-102,\) determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. $$ e^{x}=\frac{1}{\ln x} $$

Solve each equation. $$ 3^{x^{2}-12}=9^{2 x} $$

Write each equation in its equivalent exponential form. Then solve for \(x .\) $$ \log _{4} x=-3 $$

The loudness level of a sound can be expressed by comparing the sound's intensity to the intensity of a sound barely audible to the human ear. The formula $$ D-10\left(\log I-\log I_{0}\right) $$ describes the loudness level of a sound, \(D\), in decibels, where \(I\) is the intensity of the sound, in watts per meter \(^{2},\) and \(I_{0}\) is the intensity of a sound barely audible to the human ear. a. Express the formula so that the expression in parentheses is written as a single logarithm. b. Use the form of the formula from part (a) to answer this question: If a sound has an intensity 100 times the intensity of a softer sound, how much larger on the decibel scale is the loudness level of the more intense sound?

The formula \(A=37.3 e^{0.0095t}\) models the population of California, \(A,\) in millions, \(t\) years after 2010 . a. What was the population of California in \(2010 ?\) b. When will the population of California reach 40 million?

Write each equation in its equivalent exponential form. Then solve for \(x .\) $$ \log _{64} x=\frac{2}{3} $$

The formula \(t-\frac{1}{c}[\ln A-\ln (A-N)]\) describes the time, \(t,\) in weeks, that it takes to achieve mastery of a portion of a task, where \(A\) is the maximum learning possible, \(N\) is the portion of the learning that is to be achieved, and \(c\) is a constant used to measure an individual's learning style. a. Express the formula so that the expression in brackets is written as a single logarithm. b. The formula is also used to determine how long it will take chimpanzees and apes to master a task. For example, a typical chimpanzee learning sign language can master a maximum of 65 signs. Use the form of the formula from part (a) to answer this question: How many weeks will it take a chimpanzee to master 30 signs if \(c\) for that chimp is \(0.03 ?\)

The formula \(A=25.1 e^{0.0187t}\) models the population of Texas, \(A,\) in millions, \(t\) years after 2010 . a. What was the population of Texas in \(2010 ?\) b. When will the population of Texas reach 28 million?

Evaluate each expression without using a calculator. $$ \log _{3}\left(\log _{7} 7\right) $$