91Ó°ÊÓ

Use the zero or root feature or the zoom and trace features of a graphing utility to approximate the solution of the exponential equation accurate to three decimal places. $$\left(1+\frac{0.065}{365}\right)^{365 t}=4$$

(a) use a graphing utility to graph the two equations in the same viewing window and (b) use the table feature of the graphing utility to create a table of values for each equation. (c) What do the graphs and tables suggest? Verify your conclusion algebraically.$$y_{1}=2\left[\ln 8-\ln \left(x^{2}+1\right)\right], \quad y_{2}=\ln \left[\frac{64}{\left(x^{2}+1\right)^{2}}\right]$$

Use the properties of natural logarithms to rewrite the expression. $$e \ln 1$$

(a) use a graphing utility to graph the two equations in the same viewing window and (b) use the table feature of the graphing utility to create a table of values for each equation. (c) What do the graphs and tables suggest? Verify your conclusion algebraically.$$y_{1}=2\left[\ln 6+\ln \left(x^{2}+1\right)\right], \quad y_{2}=\ln \left[36\left(x^{2}+1\right)^{2}\right]$$.

Exploration Use a graphing utility to graph \(y_{1}=e^{x}\) and each of the functions \(y_{2}=x^{2}, y_{3}=x^{3}, y_{4}=\sqrt{x}\) and \(y_{5}=|x|\) in the same viewing window. (a) Which function increases at the fastest rate for "large" values of \(x ?\) (b) Use the result of part (a) to make a conjecture about the rates of growth of \(y_{1}=e^{x}\) and \(y=x^{n},\) where \(n\) is a natural number and \(x\) is "large." (c) Use the results of parts (a) and (b) to describe what is implied when it is stated that a quantity is growing exponentially.

Use the zero or root feature or the zoom and trace features of a graphing utility to approximate the solution of the exponential equation accurate to three decimal places. $$\left(4-\frac{2.471}{40}\right)^{9 t}=21$$

Use the properties of natural logarithms to rewrite the expression. $$e^{\ln 22}$$

Use the properties of natural logarithms to rewrite the expression. $$\ln e^{\ln e}$$

Think About It Graph \(y=3^{x}\) and \(y=4^{x}\). Use the graph to solve the inequality \(3^{x} < 4^{x}\)

(a) use a graphing utility to graph the two equations in the same viewing window and (b) use the table feature of the graphing utility to create a table of values for each equation. (c) What do the graphs and tables suggest? Verify your conclusion algebraically.$$y_{1}=\ln x+\frac{1}{2} \ln (x+1), \quad y_{2}=\ln (x \sqrt{x+1})$$