Chapter 4

Solve the logarithmic equation for \(x .\) $$4-\log (3-x)=3$$

Draw graphs of the given family of functions for \(c=0.25,0.5,1,2,4 .\) How are the graphs related? $$f(x)=c 2^{x}$$

Use the Laws of Logarithms to expand the expression. $$\ln \left(\frac{x^{3} \sqrt{x-1}}{3 x+4}\right)$$

Sketch the graph of the function by plotting points. $$g(x)=1+\log x$$

Solve the logarithmic equation for \(x .\) $$\log _{2}\left(x^{2}-x-2\right)=2$$

Draw graphs of the given family of functions for \(c=0.25,0.5,1,2,4 .\) How are the graphs related? $$f(x)=2^{c x}$$

Use the Laws of Logarithms to expand the expression. $$\log \left(\frac{10^{x}}{x\left(x^{2}+1\right)\left(x^{4}+2\right)}\right)$$

Solve the logarithmic equation for \(x .\) $$\log _{2} 3+\log _{2} x=\log _{2} 5+\log _{2}(x-2)$$

Find, rounded to two decimal places, (a) the intervals on which the function is increasing or decreasing and (b) the range of the function. $$y=10^{x-x^{2}}$$

Use the Laws of Logarithms to combine the expression. $$\log _{3} 5+5 \log _{3} 2$$

Solve the logarithmic equation for \(x .\) $$2 \log x=\log 2+\log (3 x-4)$$

Use the Laws of Logarithms to combine the expression. $$\log 12+\frac{1}{2} \log 7-\log 2$$

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

Bacteria Growth A bacteria culture contains 1500 bacteria initially and doubles every hour. (a) Find a function that models the number of bacteria after \(t\) hours. (b) Find the number of bacteria after 24 hours.

Use the Laws of Logarithms to combine the expression. $$\log _{2} A+\log _{2} B-2 \log _{2} C$$

Solve the logarithmic equation for \(x .\) $$\log _{5} x+\log _{5}(x+1)=\log _{5} 20$$

Mouse Population A certain breed of mouse was introduced onto a small island with an initial population of 320 mice, and scientists estimate that the mouse population is doubling every year. (a) Find a function that models the number of mice after \(t\) years. (b) Estimate the mouse population after 8 years.

Use the Laws of Logarithms to combine the expression. $$\log _{5}\left(x^{2}-1\right)-\log _{5}(x-1)$$

Solve the logarithmic equation for \(x .\) $$\log _{5}(x+1)-\log _{5}(x-1)=2$$

Compound Interest An investment of \(\$ 5000\) is deposited into an account in which interest is compounded monthly. Complete the table by filling in the amounts to which the investment grows at the indicated times or interest rates. \(r=4 \%\) $$\begin{array}{|c|c|} \hline \begin{array}{c} \text { Time } \\\\\text { (years) }\end{array} & \text { Amount } \\\\\hline 1 & \\\2 & \\\3 & \\\4 & \\\5 & \\\6 & \\\\\hline\end{array}$$

Use the Laws of Logarithms to combine the expression. $$4 \log x-\frac{1}{3} \log \left(x^{2}+1\right)+2 \log (x-1)$$

Solve the logarithmic equation for \(x .\) $$\log _{3}(x+15)-\log _{3}(x-1)=2$$

Use the Laws of Logarithms to combine the expression. $$\ln (a+b)+\ln (a-b)-2 \ln c$$

Draw the graph of \(y=4^{x},\) then use it to draw the graph of \(y=\log _{4} x\).

Solve the logarithmic equation for \(x .\) $$\log _{2} x+\log _{2}(x-3)=2$$

Compound Interest If \(\$ 10,000\) is invested at an interest rate of \(3 \%\) per year, compounded semiannually, find the value of the investment after the given number of years. (a) 5 years (b) 10 years (c) 15 years

Use the Laws of Logarithms to combine the expression. $$\ln 5+2 \ln x+3 \ln \left(x^{2}+5\right)$$

Draw the graph of \(y=3^{x},\) then use it to draw the graph of \(y=\log _{3} x\).

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

Compound Interest If \(\$ 2500\) is invested at an interest rate of \(2.5 \%\) per year, compounded daily, find the value of the investment after the given number of years. (a) 2 years (b) 3 years (c) 6 years

Use the Laws of Logarithms to combine the expression. $$2\left(\log _{5} x+2 \log _{5} y-3 \log _{5} z\right)$$

Solve the logarithmic equation for \(x .\) $$\log _{9}(x-5)+\log _{9}(x+3)=1$$

Compound Interest If \(\$ 500\) is invested at an interest rate of \(3.75 \%\) per year, compounded quarterly, find the value of the investment after the given number of years. (a) 1 year (b) 2 years (c) 10 years

Use the Laws of Logarithms to combine the expression. $$\frac{1}{3} \log (x+2)^{3}+\frac{1}{2}\left[\log x^{4}-\log \left(x^{2}-x-6\right)^{2}\right]$$

Solve the logarithmic equation for \(x .\) $$\ln (x-1)+\ln (x+2)=1$$

Compound Interest If \(\$ 4000\) is borrowed at a rate of \(5.75 \%\) interest per year, compounded quarterly, find the amount due at the end of the given number of years. (a) 4 years (b) 6 years (c) 8 years

Use the Laws of Logarithms to combine the expression. $$\log _{a} b+c \log _{a} d-r \log _{a} s$$

For what value of \(x\) is the following true? $$\log (x+3)=\log x+\log 3$$

Present Value The present value of a sum of money is the amount that must be invested now, at a given rate of interest, to produce the desired sum at a later date. Find the present value of \(\$ 10,000\) if interest is paid at a rate of \(9 \%\) per year, compounded semiannually, for 3 years.

Use the Change of Base Formula and a calculator to evaluate the logarithm, rounded to six decimal places. Use either natural or common logarithms. $$\log _{2} 5$$

For what value of \(x\) is it true that \((\log x)^{3}=3 \log x ?\)

Present Value The present value of a sum of money is the amount that must be invested now, at a given rate of interest, to produce the desired sum at a later date. Find the present value of \(\$ 100,000\) if interest is paid at a rate of \(8 \%\) per year, compounded monthly, for 5 years.

Use the Change of Base Formula and a calculator to evaluate the logarithm, rounded to six decimal places. Use either natural or common logarithms. $$\log _{5} 2$$

Solve for \(x: 2^{2 / \log _{5} x}=\frac{1}{16}\)

Present Value The present value of a sum of money is the amount that must be invested now, at a given rate of interest, to produce the desired sum at a later date. Annual Percentage Yield Find the annual percentage yield for an investment that earns \(8 \%\) per year, compounded monthly.

Use the Change of Base Formula and a calculator to evaluate the logarithm, rounded to six decimal places. Use either natural or common logarithms. $$\log _{3} 16$$

Solve for \(x: \) $$\log _{2}\left(\log _{3} x\right)=4$$

Use the Change of Base Formula and a calculator to evaluate the logarithm, rounded to six decimal places. Use either natural or common logarithms. $$\log _{6} 92$$

Use a graphing device to find all solutions of the equation, rounded to two decimal places. $$\ln x=3-x$$

Growth of an Exponential Function Suppose you are offered a job that lasts one month, and you are to be very well paid. Which of the following methods of payment is more profitable for you? (a) One million dollars at the end of the month (b) Two cents on the first day of the month, 4 cents on the second day, 8 cents on the third day, and, in general, \(2^{n}\) cents on the \(n\) th day