Chapter 5

In general, it is not possible to find exact solutions analytically for equations that involve exponential or logarithmic functions together with polynomial, radical, and rational functions. Solve each equation= using a graphical method, and express solutions to the nearest thousandth if an approximation is appropriate. $$e^{x}=\frac{1}{x+2}$$

The number of species in a sample is approximated by \(S(n)=a \ln \left(1+\frac{n}{a}\right),\) where \(n\) is the number of individuals in the sample and \(a\) is a constant that indicates the diversity of species in the community. If \(a=0.36,\) find \(S(n)\) for each value of \(n\) (Hint: \(S(n)\) must be a whole number.) (a) 100 (b) 200 (c) 150 (d) 10

In general, it is not possible to find exact solutions analytically for equations that involve exponential or logarithmic functions together with polynomial, radical, and rational functions. Solve each equation= using a graphical method, and express solutions to the nearest thousandth if an approximation is appropriate. $$3^{-x}=\sqrt{x+5}$$

Use a graph with the given viewing window to decide which functions are one- to-one. If a function is one-to-one, give the equation of its inverse function. Check your work by graphing the inverse function on the same coordinate axes. $$f(x)=x^{4}-5 x^{2}+6 ;[-3,3] \text { by }[-1,8]$$

(a) Prove the quotient rule of logarithms. $$\log _{a} \frac{x}{y}=\log _{a} x-\log _{a} y$$ (b) Prove the power rule of logarithms. $$\log _{a} x^{r}=r \log _{a} x$$

Use any method (analytic or graphical) to solve each equation. $$\log _{2} \sqrt{2 x^{2}}-1=0.5$$

Use a graph with the given viewing window to decide which functions are one- to-one. If a function is one-to-one, give the equation of its inverse function. Check your work by graphing the inverse function on the same coordinate axes. $$f(x)=\frac{x-5}{x+3} ;[-9.4,9.4] \text { by }[-6.2,6.2]$$

Use any method (analytic or graphical) to solve each equation. $$\log x^{2}=(\log x)^{2}$$

Use any method (analytic or graphical) to solve each equation. $$\ln \left(\ln e^{-x}\right)=\ln 3$$

Use any method (analytic or graphical) to solve each equation. $$e^{x+\ln 3}=4 e^{x}$$

surface of the ocean due to rapid evaporation. In the higher latitudes, there is less evaporation, and rainfall causes the salinity to be less on the surface than at lower depths. The function given by $$ f(x)=31.5+1.1 \log (x+1) $$ models salinity to depths of 1000 meters at a latitude of \(57.5^{\circ} \mathrm{N} .\) The variable \(x\) is the depth in meters, and \(f(x)\) is in grams of salt per kilogram of seawater. (Source: Hartman, \(D\), Global Physical Climatology, Academic Press.) Approximate analytically the depth where the salinity equals 33

Restrict the domain so that the function is one-to-one and the range is not changed. You may wish to use a graph to help decide. Answers may vary. $$f(x)=-x^{2}+4$$

Restrict the domain so that the function is one-to-one and the range is not changed. You may wish to use a graph to help decide. Answers may vary. $$f(x)=(x-1)^{2}$$

surface of the ocean due to rapid evaporation. In the higher latitudes, there is less evaporation, and rainfall causes the salinity to be less on the surface than at lower depths. The function given by $$ f(x)=31.5+1.1 \log (x+1) $$ models salinity to depths of 1000 meters at a latitude of \(57.5^{\circ} \mathrm{N} .\) The variable \(x\) is the depth in meters, and \(f(x)\) is in grams of salt per kilogram of seawater. (Source: Hartman, \(D\), Global Physical Climatology, Academic Press.) Estimate the salinity at a depth of 500 meters.

Restrict the domain so that the function is one-to-one and the range is not changed. You may wish to use a graph to help decide. Answers may vary. $$f(x)=|x-6|$$

Restrict the domain so that the function is one-to-one and the range is not changed. You may wish to use a graph to help decide. Answers may vary. $$f(x)=x^{4}$$

Restrict the domain so that the function is one-to-one and the range is not changed. You may wish to use a graph to help decide. Answers may vary. $$f(x)=-\sqrt{x^{2}-16}$$

Using the restrictions on the functions in Exercises \(123-126,\) find a formula for \(f^{-1}\). $$f(x)=-x^{2}+4, \quad x \geq 0$$