Chapter 5

$$\text { The given equations are quadratic in form. Solve each and give exact solutions.}$$ $$\frac{1}{2} e^{2 x}+e^{x}=1$$

Assume that \(f(x)=a^{x},\) where \(a>1\) If the point \((p, q)\) is on the graph of \(f,\) then the point ______ is on the graph of \(f^{-1}\)

Use the properties of logarithms to rewrite each expression as a single logarithm with coefficient 1. Assume that all variables represent positive real numbers. $$\frac{1}{2} \log x-\frac{1}{3} \log y-2 \log z$$

$$\text { The given equations are quadratic in form. Solve each and give exact solutions.}$$ $$\frac{1}{4} e^{2 x}+2 e^{x}=3$$

Use the properties of logarithms to rewrite each expression as a single logarithm with coefficient 1. Assume that all variables represent positive real numbers. $$\ln (a+b)+\ln a-\frac{1}{2} \ln 4$$

$$\text { The given equations are quadratic in form. Solve each and give exact solutions.}$$ $$3^{2 x}+35=12\left(3^{x}\right)$$

Use the properties of logarithms to rewrite each expression as a single logarithm with coefficient 1. Assume that all variables represent positive real numbers. $$\frac{4}{3} \ln m-\frac{2}{3} \ln 8 n-\ln m^{3} n^{2}$$

$$\text { The given equations are quadratic in form. Solve each and give exact solutions.}$$ $$5^{2 x}+3\left(5^{x}\right)=28$$

Use the change-of-base rule to find an approximation for each logarithm. $$\log _{5} 10$$

$$\text { The given equations are quadratic in form. Solve each and give exact solutions.}$$ $$\left(\log _{2} x\right)^{2}+\log _{2} x=2$$

Use the change-of-base rule to find an approximation for each logarithm. $$\log _{9} 12$$

$$\text { The given equations are quadratic in form. Solve each and give exact solutions.}$$ $$(\log x)^{2}-6 \log x=7$$

Use the change-of-base rule to find an approximation for each logarithm. $$\log _{15} 5$$

$$\text { The given equations are quadratic in form. Solve each and give exact solutions.}$$ $$(\ln x)^{2}+16=10 \ln x$$

Use the change-of-base rule to find an approximation for each logarithm. $$\log _{1 / 2} 3$$

$$\text { The given equations are quadratic in form. Solve each and give exact solutions.}$$ $$2(\ln x)^{2}+9 \ln x=5$$

Use the change-of-base rule to find an approximation for each logarithm. $$\log _{100} 83$$

For the given \(f(x)\), solve the equation \(f(x)=0\) analytically and then use a graph of \(y=f(x)\) to solve the inequalities \(f(x)<0\) and \(f(x) \geq 0\) $$f(x)=-2 e^{x}+5$$

Use the change-of-base rule to find an approximation for each logarithm. $$\log _{200} 175$$

Decide whether the pair of functions \(f\) and \(g\) are inverses. Assume axes have equal scales. CAN'T COPY THE GRAPH $$f(x)=-\frac{3}{11} x, \quad g(x)=-\frac{11}{3} x$$

For the given \(f(x)\), solve the equation \(f(x)=0\) analytically and then use a graph of \(y=f(x)\) to solve the inequalities \(f(x)<0\) and \(f(x) \geq 0\) $$f(x)=-3 e^{x}+7$$

Use the change-of-base rule to find an approximation for each logarithm. $$\log _{29} 7.5$$

Decide whether the pair of functions \(f\) and \(g\) are inverses. Assume axes have equal scales. CAN'T COPY THE GRAPH $$f(x)=2 x+4, \quad g(x)=\frac{1}{2} x-2$$

For the given \(f(x)\), solve the equation \(f(x)=0\) analytically and then use a graph of \(y=f(x)\) to solve the inequalities \(f(x)<0\) and \(f(x) \geq 0\) $$f(x)=2\left(3^{x}\right)-18$$

Use the change-of-base rule to find an approximation for each logarithm. $$\log _{5.8} 12.7$$

Decide whether the pair of functions \(f\) and \(g\) are inverses. Assume axes have equal scales. CAN'T COPY THE GRAPH $$f(x)=5 x-5, \quad g(x)=\frac{1}{5} x+5$$

For the given \(f(x)\), solve the equation \(f(x)=0\) analytically and then use a graph of \(y=f(x)\) to solve the inequalities \(f(x)<0\) and \(f(x) \geq 0\) $$f(x)=4^{x-2}-2$$

For the given \(f(x)\), solve the equation \(f(x)=0\) analytically and then use a graph of \(y=f(x)\) to solve the inequalities \(f(x)<0\) and \(f(x) \geq 0\) $$f(x)=3^{2 x}-9^{x+1}$$

For the given \(f(x)\), solve the equation \(f(x)=0\) analytically and then use a graph of \(y=f(x)\) to solve the inequalities \(f(x)<0\) and \(f(x) \geq 0\) $$f(x)=2^{3 x}-8^{x-3}$$

For the given \(f(x)\), solve the equation \(f(x)=0\) analytically and then use a graph of \(y=f(x)\) to solve the inequalities \(f(x)<0\) and \(f(x) \geq 0\) $$f(x)=8-4 \log _{5} x$$

For individual or group investigation (Exercises \(97-102\) ) Work Exercises \(97-102\) in order. Solve \(0=-3^{x}+7\) for \(x,\) expressing \(x\) in terms of base 3 logarithm.

For the given \(f(x)\), solve the equation \(f(x)=0\) analytically and then use a graph of \(y=f(x)\) to solve the inequalities \(f(x)<0\) and \(f(x) \geq 0\) $$f(x)=9 \log _{3}(3 x)-18$$

For the given \(f(x)\), solve the equation \(f(x)=0\) analytically and then use a graph of \(y=f(x)\) to solve the inequalities \(f(x)<0\) and \(f(x) \geq 0\) $$f(x)=\ln (x+2)$$

For the given \(f(x)\), solve the equation \(f(x)=0\) analytically and then use a graph of \(y=f(x)\) to solve the inequalities \(f(x)<0\) and \(f(x) \geq 0\) $$f(x)=\ln (x-1)-\ln (x+1)$$

Graph the inverse of each one-to-one function. CAN'T COPY THE GRAPH

For the given \(f(x)\), solve the equation \(f(x)=0\) analytically and then use a graph of \(y=f(x)\) to solve the inequalities \(f(x)<0\) and \(f(x) \geq 0\) $$f(x)=7-5 \log x$$

The equations are identities because they are true for all real numbers. Use properties of logarithms to simplify the expression on the left side of the equation so that it equals the expression on the right side, where \(x\) is any real number. $$\ln |x+\sqrt{x^{2}+3}|+\ln |x-\sqrt{x^{2}+3}|=\ln 3$$

For the given \(f(x)\), solve the equation \(f(x)=0\) analytically and then use a graph of \(y=f(x)\) to solve the inequalities \(f(x)<0\) and \(f(x) \geq 0\) $$f(x)=3-2 \log _{4}(x-5)$$

The equations are identities because they are true for all real numbers. Use properties of logarithms to simplify the expression on the left side of the equation so that it equals the expression on the right side, where \(x\) is any real number. $$\ln \left|x^{2}-\sqrt{x^{4}+1}\right|+\ln \left|x^{2}+\sqrt{x^{4}+1}\right|=0$$

Suppose \(f(x)\) is the number of cars that can be built for \(x\) dollars. What does \(f^{-1}(1000)\) represent?

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. $$x^{2}=2^{x}$$

The equations are identities because they are true for all real numbers. Use properties of logarithms to simplify the expression on the left side of the equation so that it equals the expression on the right side, where \(x\) is any real number. $$\frac{1}{3} \ln \left(\frac{x^{2}+1}{5}\right)-\frac{1}{3} \ln \left(\frac{x^{2}+4}{5}\right)=\ln \sqrt[3]{\frac{x^{2}+1}{x^{2}+4}}$$

Suppose \(f(r)\) is the volume (in cubic inches) of a sphere of radius \(r\) inches. What does \(f^{-1}(5)\) represent?

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. $$x^{2}-4=e^{x-4}+4$$

The equations are identities because they are true for all real numbers. Use properties of logarithms to simplify the expression on the left side of the equation so that it equals the expression on the right side, where \(x\) is any real number. $$\frac{1}{2} \ln \left(\frac{x^{2}}{7}\right)-\frac{1}{2} \ln \left(\frac{x^{4}+x^{2}}{7}\right)=\ln \sqrt{\frac{1}{x^{2}+1}}$$

If a line has nonzero slope \(a\), what is the slope of its reflection across the line \(y=x ?\)

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. $$\log x=x^{2}-8 x+14$$

Suppose a sample of a small community shows two species with 50 individuals each. Find the index of diversity \(H=-\left(P_{1} \log _{2} P_{1}+P_{2} \log _{2} P_{2}+\cdots+P_{n} \log _{2} P_{n}\right)\).

$$\text { Find } f^{-1}(f(2)), \text { where } f(2)=3$$

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. $$\ln x=-\sqrt[3]{x+3}$$