Chapter 13

In Problems 7 through 32, solve for \(x .\) $$ \frac{4}{\ln (x+1)}+5=13 $$

In Problems 7 through 32, solve for \(x .\) $$ \left[\frac{3}{\ln (2 x+1)}\right]^{2}-1=10 $$

In Problems 7 through 32, solve for \(x .\) $$ \frac{3}{\left(e^{x}+1\right)^{2}}=27 $$

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

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

In Problems 33 throuogh 36, solve for \(x ; O, R\), and \(S\) are positive constants. (a) \(3^{5 x+2}=100\) (b) \(Q^{2 x+1}=R\)

In Problems 33 throuogh 36, solve for \(x ; O, R\), and \(S\) are positive constants. (a) \(2 Q^{x+5}=R\) (b) \((2 Q)^{x+5}=R\)

In Problems 33 throuogh 36, solve for \(x ; O, R\), and \(S\) are positive constants. (a) \((Q+R)^{x}=S\) (b) \((O R)^{x}=S\)

In Problems 33 throuogh 36, solve for \(x ; O, R\), and \(S\) are positive constants. (a) \(R^{-x+1}-Q=S\) (b) \(\frac{R}{R^{x}}-S=R\)

Acidity is determined by the concentration of hydrogen ions in a solution. The pH scale, proposed by Sorensen in the early \(1900 \mathrm{~s}\), defines \(\mathrm{pH}\) to be \(-\log \left[H^{+}\right]\), where \(\left[H^{+}\right]\) is the concentration of hydrogen ions given in moles per liter. A pH of 7 is considered neutral; a pH greater than 7 means the solution is basic, while a pH of less than 7 indicates acidity. (a) If the concentration of hydrogen ions in a solution is increased tenfold, what happens to the \(\mathrm{pH}\) ? (b) If a blood sample has a hydrogen ion concentration of \(3.15 \times 10^{-8}\), what is the \(\mathrm{pH} ?\) (c) You'll find that the blood sample described in part (b) is mildly basic. Which has a higher concentration of hydrogen ions: the blood sample or something neutral? How many times greater is it?

In Problems 38 through 44 find all \(x\) for which each equation is true. $$ [\log x]^{3}=\log \left(x^{3}\right) $$

In Problems 38 through 44 find all \(x\) for which each equation is true. $$ e^{x^{3}}=\left(e^{x}\right)^{3} $$

In Problems 38 through 44 find all \(x\) for which each equation is true. $$ \ln x^{-1}=\frac{1}{\ln x} $$

In Problems 38 through 44 find all \(x\) for which each equation is true. $$ \frac{\ln x}{\ln 2}=\ln x-\ln 2 $$

In Problems 38 through 44 find all \(x\) for which each equation is true. $$ e^{x+1}=e^{x}+e^{1} $$

In Problems 38 through 44 find all \(x\) for which each equation is true. $$ 10^{2 x}=10^{2} 10^{x} $$

In Problems 38 through 44 find all \(x\) for which each equation is true. $$ \sqrt{\ln x}=\frac{1}{2} \ln x $$

Suppose \(\$ M_{0}\) is put in a bank account where it grows according to: \(M(t)=M_{0}\left(1+\frac{r}{12}\right)^{12 t}\), where \(t\) is in years. (a) If \(r=0.05\), how long will it take for the amount of money in the account to increase by \(50 \%\) ? (b) If the money doubles in exactly 8 years, what is \(r\) ?

Find the equation of the straight line through the points \((2, \ln 2)\) and \((3, \ln 3)\).

Find the equation of the line through the points \((2, \ln 2)\) and \((2+\epsilon, \ln (2+\epsilon))\).

The "Rule of \(70^{\text {"' }}\) says that if a quantity grows exponentially at a rate of \(r \%\) per unit of time, then its doubling time is usually about \(70 / r .\) This is merely a rule of thumb. Now we will determine how accurate an estimate this is and for what values of \(r\) it should be applied. Suppose that a quantity \(Q\) grows exponentially at \(r \%\) per unit of time \(t .\) Thus, \(Q(t)=Q_{0}\left(1+\frac{r}{100}\right)^{t}\) (a) Let \(D(r)\) be the doubling time of \(Q\) as a function of \(r .\) Find an equation for \(D(r)\). (b) On your graphing calculator, graph \(D(r)\) and \(70 / r .\) Take note of the values of \(r\) for which the latter is a good approximation of the former.