Chapter 13

Sketch the graphs of \(f(x)=2^{x}\) and the graph of \(\log _{2} x\) on the same set of axes. Label three points on each graph.

Simplify the following: \(\log x\) is shorthand for \(\log _{10} x\). (a) \(3^{\log _{3} 2}\) (b) \(\log x+\log x^{2}-3 \log x\) (c) \(2 \log (x+3)-3 \log (x+3)+\log \left(10^{\sqrt{7}}\right)\) (d) \(10^{\log x^{2}}\) (e) \(10^{3 \log x}\) (f) \(10^{-\log x}\) (g) \(10^{-0.5 \log x}\) (h) \(3^{-\log _{3}(x+y)}\) (i) \(2^{\left(\log _{2} 10-\log _{2} 5\right)}\) (j) \(10^{\frac{\log x}{2}}\)

Sketch the graph of the function without the use of a computer or graphing calculator. $$ y=\ln (x+1) $$

Fill in the blanks: When we write \(\log _{2} 3\), we say "log base two of \(3 .\) " We mean the power to which 2 must be raised in order to get 3 . (a) When we write \(\log _{5} 14\), we say " must be raised in order to get (b) When we say "log base 4 of 8 ," we write must be raised in order to get (c) We mean the power to which \(e\) must be raised in order to get 5 , so we write and we say

If \(\log _{2} u=A\) and \(\log _{2} w=B\), express the following in terms of \(A\) and \(B\) (eliminating \(u\) and \(w\) ). (a) \(\log _{2}\left(u^{2} w\right)\) (b) \(\log _{2}\left(u^{3} / w^{2}\right)\) (c) \(\log _{2}(1 / \sqrt{w})\) (d) \(\log _{2}\left(\frac{2}{\sqrt{u w}}\right)\)

Solve for \(x:\) (Don't expect "pretty" answers.) (a) \(10^{2 x}=93\) (b) \(10^{3 x+2}=1,000,000\) (c) \(2^{x+1}=7\) (d) \(3^{x} 3^{x^{2}}=3\) (e) \(5 B^{x}=(2 C)^{x+1}\) (f) \(\ln x+2=5\) (g) \(\log _{10} x=17\) (h) \(\ln (5 x-40)=3\) (i) \(\log _{10}\left(2 x^{2}+4\right)=2\) (j) \(3 \cdot 2^{x / 7}-4=12\)

Sketch the graph of the function without the use of a computer or graphing calculator. $$ y=\ln \left(x^{2}\right) $$

Approximate the values of the logarithms by giving two consecutive integers, one of which is a lower bound and the other an upper bound for the expressions given. Do this without a calculator. (You can use the calculator to check your answers, but the idea of the problem is to get you to think about what logarithms mean.) Explain your reasoning as in the example below. (a) \(\log _{7} 50\) (b) \(\log _{10}(0.5)\)

For Problems 3 through 9 , simplify the expression given. (a) \(\sqrt{2} \cot 10^{\log 7}\) (b) \(\pi e^{\ln 4}\)

(a) Approximate \(\log _{3} 16\) (with error less than \(0.005\) ) using your calculator. (b) Rewrite \(\log _{3} 16\) in terms of \(\log\) base 10 . (c) Rewrite \(\log _{3} 16\) in terms of log base \(e\). (d) Rewrite \(\log _{3} 16\) in terms of log base 7 .

Sketch the graph of the function without the use of a computer or graphing calculator. $$ y=|\ln x| $$

Approximate the values of the logarithms by giving two consecutive integers, one of which is a lower bound and the other an upper bound for the expressions given. Do this without a calculator. (You can use the calculator to check your answers, but the idea of the problem is to get you to think about what logarithms mean.) Explain your reasoning as in the example below. (a) \(\log _{10}(0.05)\) (b) \(\log _{3} 29\)

For Problems 3 through 9 , simplify the expression given. (a) \(3^{2} 10^{2 \log 5}\) (b) \(5 e^{-3 \ln 2}\)

Solve for \(x\). (a) \(3 \ln x+5=(\ln x) \ln 2\) (b) \(2\left(7^{1+\log x}\right)=8\) (c) \(K e^{x}+K=L e^{x}-L\), where \(K\) and \(L\) are constants and \(0

Sketch the graph of the function without the use of a computer or graphing calculator. $$ y=\ln |x| $$

For Problems 3 through 9 , simplify the expression given. (a) \(10^{\log 2+1}\) (b) \(e^{3-\ln 2}\)

Solve for \(x\). (a) \(2^{x^{2}} 2^{x}=3^{x}\) (b) \(3^{x^{2}+2 x}=1\) (c) \(3 \ln \left(x^{4}\right)-2 \ln 2 x=10\) (d) \(e^{2 x}+e^{x}-6=0\) (e) \(e^{x}+8 e^{-x}=6\) (f) \((\ln x)(\ln 5)=\ln 4 x\)

Sketch the graph of the function without the use of a computer or graphing calculator. $$ y=\ln \left(\frac{1}{x}\right) $$

Simplify the following. (No calculators, except to check your answers if you like.) (a) \(\log _{2} \sqrt{8}\) (b) \(\log _{10} 0.001\) (c) \(\log _{2}\left(\frac{4}{\sqrt{8}}\right)\) (d) \(\log _{3}(1 / 9)\) (e) \(\log _{k} k^{3 x}\) (f) \(\log _{k} 1\) (g) \(\log _{k}\left(k^{x} k^{y}\right)\)

For Problems 3 through 9 , simplify the expression given. (a) \(2^{\log _{2} 3+3}\) (b) \(e^{2 \ln A+1}\)

The Richter scale, introduced in the mid- 1900 s, measures the intensity of earthquakes. A measurement on the Richter scale is given by $$ M=\log \frac{I}{S} $$ where \(I\) is the intensity of the quake and \(\mathrm{S}\) is some standard. Suppose we want to compare the intensity, \(I_{1}\), of a particular earthquake with the intensity, \(I_{2}\), of a less violent quake. The difference in their measurements on the Richter scale is $$ \log \frac{I_{1}}{\mathrm{~S}}-\log \frac{I_{2}}{\mathrm{~S}}=\log \left[\frac{\frac{I_{1}}{\mathrm{~S}}}{\frac{I_{2}}{\mathrm{~S}}}\right]=\log \frac{I_{1}}{I_{2}} $$ In particular, suppose that one earthquake measures 7 on the Richter scale and another measures 4 . Then $$ \log \frac{I_{1}}{I_{2}}=7-4=3 $$ Therefore, \(\frac{I_{1}}{I_{2}}=10^{3}=1000 .\) The former earthquake has 1000 times the intensity of the latter. (a) On August 20,1999 , there was an earthquake in Costa Rica ( 50 miles south of San Jose) measuring \(6.7\) on the Richter scale and another in Montana (near the Idaho border) measuring 5 on the Richter scale. How many times more intense was the Costa Rican earthquake? (b) The 1989 earthquake in San Francisco measured \(7.1\) on the Richter scale. How many times more intense was the earthquake in Turkey on August 17, 1999 , measuring \(7.4\) on the Richter scale?

Sketch a rough graph of \(y=\ln x-\ln \left(x^{3}\right)+4 \ln \left(x^{2}\right) .\) (Hint: This will be straightforward after you have rewritten in the form \(y=K \ln x\), where \(K\) is a constant.)

For Problems 3 through 9 , simplify the expression given. (a) \(10^{\log 2-\log 3}\) (b) \(e^{2 \ln 5-\ln 2}\)

In Problems 7 through 32, solve for \(x .\) $$ 5^{3 x+2}=2^{5 x} $$

For Problems 3 through 9 , simplify the expression given. (a) \(10^{-\log \frac{1}{10}}\) (b) \(e^{\frac{\ln 3}{2}}\)

In Problems 7 through 32, solve for \(x .\) $$ \frac{3}{2^{x-3}}=7^{2 x+1} $$

For Problems 3 through 9 , simplify the expression given. (a) \(10^{\frac{\log 8+1}{2}}\) (b) \(e^{-\frac{\ln 8}{3}+2}\)

In Problems 7 through 32, solve for \(x .\) $$ \pi \cdot 3^{1+2 x}=\sqrt{\pi} 5^{x} $$

In Problems 10 through 13, let \(\log 2=a\) and \(\log 3=b .\) Express each of the following in terms of a and \(b .\) There should be no logarithms explicitly in the expressions you give. $$ 5 \log \frac{2}{3} $$

In Problems 7 through 32, solve for \(x .\) $$ e^{3 x}=\left(\frac{5}{e}\right)^{x+1} $$

In Problems 7 through 32, solve for \(x .\) $$ e^{2} e^{x}=\pi^{3 x+3} $$

In Problems 10 through 13, let \(\log 2=a\) and \(\log 3=b .\) Express each of the following in terms of a and \(b .\) There should be no logarithms explicitly in the expressions you give. $$ 5 \log \sqrt[3]{6} $$

In Problems 7 through 32, solve for \(x .\) $$ 3^{x} \cdot \frac{5}{3^{x+1}}=0 $$

In Problems 7 through 32, solve for \(x .\) $$ \frac{7+\pi 3^{x+2}}{2}=3 \pi $$

(a) Evaluate the following limits. To do so rigorously, it is useful to apply L'Hôpital's rule (Appendix F). Otherwise, use a calculator to guess the answers. i. \(\lim _{x \rightarrow \infty} \frac{\sqrt{x}}{\ln x}\) ii. \(\lim _{x \rightarrow \infty} \frac{\ln x}{\sqrt{x}}\) (b) Which grows faster as \(x \rightarrow \infty, \ln x\) or \(\sqrt{x}\) ?

In Problems 7 through 32, solve for \(x .\) $$ \log x-\log (x+1)=2 $$

In Problems 7 through 32, solve for \(x .\) $$ \ln x^{2}=3+\ln x $$

In Problems 16 and 17, rewrite the expression given as a single logarithm. $$ a \ln (x+3)-b \ln \left(\frac{1}{x}\right)-c \ln (x+1) $$

In Problems 7 through 32, solve for \(x .\) $$ \ln \sqrt{x}+\ln x^{2}=1-2 \ln x $$

In Problems 7 through 32, solve for \(x .\) $$ [\ln (2 x+3)]^{2}-9=0 $$

In Problems 7 through 32, solve for \(x .\) $$ \log x[\log (x+3)-2]=0 $$

In Problems 7 through 32, solve for \(x .\) $$ e^{x}\left(e^{x}-5\right)=0 $$

In Problems 7 through 32, solve for \(x .\) $$ e^{x}\left(e^{x}-5\right)=6 $$

In Problems 7 through 32, solve for \(x .\) $$ e^{2 x}-4 e^{x}+3=0 $$

In Problems 7 through 32, solve for \(x .\) $$ 2 e^{2 x}+6=11 e^{x} $$

In Problems 7 through 32, solve for \(x .\) $$ e^{-2 x}-e^{-x}=6 $$

In Problems 7 through 32, solve for \(x .\) $$ e^{-2 x}=2 $$

In Problems 7 through 32, solve for \(x .\) $$ e^{x}-1=e^{-x} $$

In Problems 7 through 32, solve for \(x .\) $$ e^{x}-2=\frac{3}{e^{x}} $$

In Problems 7 through 32, solve for \(x .\) $$ 3^{\ln x}=5 x $$