Chapter 5

Find the difference quotient of the function. $$f(x)=10^{x}$$

Evaluate the given expression without using a calculator. $$\ln e^{x+y}$$

Simplify the expression without using a calculator. $$\frac{18-\sqrt{126}}{3}$$

(a) Solve \(7^{x}=3,\) using natural logarithms. Leave your answer in logarithmic form; don't approximate with a calculator. (b) Solve \(7^{x}=3,\) using common (base 10 ) logarithms. Leave your answer in logarithmic form. (c) Use the change of base formula in Special Topics \(5.4 . \mathrm{A}\) to show that your answers in parts (a) and (b) are the same.

Deal with the energy intensity i of a sound, which is related to the loudness of the sound by the function \(L(i)=10 \cdot \log \left(i / i_{0}\right),\) where \(i_{0}\) is the minimum intensity detectable by the human ear and \(L(i)\) is measured in decibels. Find the decibel measure of the sound. Loud conversation (intensity is 4 million times \(i_{0}\) ).

Evaluate the given expression without using a calculator. $$\ln e^{x^{2}+2 y}$$

Simplify the expression without using a calculator. $$\sqrt{50}-\sqrt{72}$$

Deal with the energy intensity i of a sound, which is related to the loudness of the sound by the function \(L(i)=10 \cdot \log \left(i / i_{0}\right),\) where \(i_{0}\) is the minimum intensity detectable by the human ear and \(L(i)\) is measured in decibels. Find the decibel measure of the sound. Victoria Falls in Africa (intensity is 10 billion times \(i_{0}\) ).

Evaluate the given expression without using a calculator. $$e^{\ln x^{2}}$$

Simplify the expression without using a calculator. $$\sqrt{150}+\sqrt{24}$$

Find the difference quotient of the function. $$f(x)=e^{x}-e^{-x}$$

Evaluate the given expression without using a calculator. $$e^{\ln (\ln 2)}$$

Simplify the expression without using a calculator. $$5 \sqrt{20}-\sqrt{45}+2 \sqrt{80}$$

The perceived loudness \(L\) of a sound of intensity \(I\) is given by \(L=k \cdot \ln I,\) where \(k\) is a certain constant. By how much must the intensity be increased to double the loudness? (That is, what must be done to \(I\) to produce \(2 L ?\) )

Find a viewing window (or windows) that shows a complete graph of the function. $$k(x)=e^{-x}$$

Write the rule of the function in the form \(\left.f(x)=P e^{k x} . \text { (See the discussion and box after Example } 11 .\right)\) $$f(x)=4\left(25^{x}\right)$$

Simplify the expression without using a calculator. $$\sqrt[3]{40}+2 \sqrt[3]{135}-5 \sqrt[3]{320}$$

Solve the equation as in Example \(8 .\) $$\ln (2 x-1)-\ln 2=\ln (3 x+6)-\ln 6$$

Compute each of the following pairs of numbers. (a) \(\log 18\) and \(\frac{\ln 18}{\ln 10}\) (b) \(\log 456\) and \(\frac{\ln 456}{\ln 10}\) (c) \(\log 8950\) and \(\frac{\ln 8950}{\ln 10}\) (d) What do these results suggest?

Write the rule of the function in the form \(\left.f(x)=P e^{k x} . \text { (See the discussion and box after Example } 11 .\right)\) $$g(x)=3.9\left(1.03^{x}\right)$$

Simplify the expression without using a calculator. $$\sqrt{16 a^{8} b^{-2}}$$

Prove that for any positive number \(c, \log c=\frac{\ln c}{\ln 10} .[\)Hint: We know that \(10^{\log c}=c\) (why?). Take natural logarithms on both sides and use a logarithm law to simplify and solve for log \(c .]\)

Write the rule of the function in the form \(\left.f(x)=P e^{k x} . \text { (See the discussion and box after Example } 11 .\right)\) $$g(x)=-16\left(30.5^{x}\right)$$

Simplify the expression without using a calculator. $$\sqrt{54 m^{-6} n^{3}}$$

Find each of the following logarithms. (a) \(\log 8.753\) (b) \(\log 87.53\) (c) \(\log 875.3\) (d) \(\log 8753\) (e) \(\log 87,530\) (f) How are the numbers \(8.753,87.53, \ldots, 87,530\) related to one another? How are their logarithms related? State a general conclusion that this evidence suggests.

Write the rule of the function in the form \(\left.f(x)=P e^{k x} . \text { (See the discussion and box after Example } 11 .\right)\) $$f(x)=-2.2\left(.75^{x}\right)$$

Simplify the expression without using a calculator. $$\frac{\sqrt{c^{2} d^{6}}}{\sqrt{4 c^{3} d^{-4}}}$$

Prove that for every positive number \(c, \log c\) can be written in the form \(k+\log b,\) where \(k\) is an integer and \(1 \leq b<10 .\) [Hint: Write \(c\) in scientific notation and use logarithm laws to express log \(c \text { in the required form. }]\)

Simplify the expression without using a calculator. $$\frac{\sqrt{a^{-10} b^{-12}}}{\sqrt{a^{14} d^{-4}}}$$

Write the rule of the function in the form \(\left.f(x)=a^{x} . \text { (See the discussion and box after Example } 11 .\right)\) $$f(x)=e^{1.6094 x}$$

Simplify the expression without using a calculator. $$\frac{\sqrt[3]{a^{5} b^{4} c^{3}}}{\sqrt[3]{a^{-1} b^{2} c^{6}}}$$

Wayland and Christy have been tracking the number of cases of flu in their city: $$\begin{array}{|l|c|c|c|c|c|c|c|}\hline \text { Weeks since January 1 } & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\\\\hline \text { Number of cases } & 10 & 13 & 16 & 20 & 24 & 31 & 38 \\\\\hline\end{array}$$ Wayland thinks this is exponential growth. Christy doesn't think so. After playing around with the data, they plot the points and still disagree. (a) Plot the points. Do you agree with Wayland or with Christy? (b) They create a new plot, this time using the natural logarithms of the number of cases. So they plot the points \((0, \ln (10)),(2, \ln (13)),\) etc. As soon as they see this new plot, they agree! Construct this new plot. (c) Who was right, Wayland or Christy? Why?

Find the domain of the given function (that is, the largest set of real numbers for which the rule produces well-defined real numbers). $$f(x)=\ln (x+1)$$

Simplify the expression without using a calculator. $$\frac{\sqrt[5]{16 a^{4} b^{2}}}{\sqrt[5]{2^{-1} a^{14} b^{-3}}}$$

Find the domain of the given function (that is, the largest set of real numbers for which the rule produces well-defined real numbers). $$g(x)=\ln (x+2)$$

Rationalize the denominator and simplify your answer. $$\frac{3}{\sqrt{8}}$$

Solve the equation. $$\ln (x+9)-\ln x=1$$

Find the domain of the given function (that is, the largest set of real numbers for which the rule produces well-defined real numbers). $$h(x)=\log (-x)$$

Rationalize the denominator and simplify your answer. $$\frac{2}{\sqrt{6}}$$

Solve the equation. $$\ln (3 x+5)-1=\ln (2 x-3)$$

List all asymptotes of the graph of the function and the approximate coordinates of each local extremum. $$g(x)=x 2^{-x}$$

Rationalize the denominator and simplify your answer. $$\frac{3}{2+\sqrt{12}}$$

Solve the equation. $$\log x+\log (x-3)=1$$

List all asymptotes of the graph of the function and the approximate coordinates of each local extremum. $$h(x)=e^{x^{2} / 2}$$

(a) Graph \(y=x\) and \(y=e^{\ln x}\) in separate viewing windows [or use a split- screen if your calculator has that feature]. For what values of \(x\) are the graphs identical? (b) Use the properties of logarithms to explain your answer in part (a).

Rationalize the denominator and simplify your answer. $$\frac{1+\sqrt{3}}{5+\sqrt{10}}$$

Solve the equation. $$\log (x-4)+\log (x-1)=1$$

List all asymptotes of the graph of the function and the approximate coordinates of each local extremum. $$k(x)=2^{x^{2}-6 x+2}$$

(a) Graph \(y=x\) and \(y=\ln \left(e^{x}\right)\) in separate viewing windows [or a split-screen if your calculator has that feature]. For what values of \(x\) are the graphs identical? (b) Use the properties of logarithms to explain your answer in part (a).

Rationalize the denominator and simplify your answer. $$\begin{aligned} &1\\\ &\frac{2}{\sqrt{x}+2} \end{aligned}$$