Chapter 4

Solve the exponential equation algebraically. Approximate the result to three decimal places.\(6^{x}+10=47\)

Evaluate the logarithm. Round your result to three decimal places.\(\log _{19} 42\)

Evaluate the expression without using a calculator.\(\log _{5} 125\)

Sketch the graph of the function.\(y=3^{-x^{2}}\)

Solve the exponential equation algebraically. Approximate the result to three decimal places.\(3^{2 x}=80\)

Evaluate the logarithm. Round your result to three decimal places.\(\log _{15} 1250\)

Evaluate the expression without using a calculator.\(\log _{2} \frac{1}{16}\)

Sketch the graph of the function.\(y=e^{-0.1 x}\)

Solve the exponential equation algebraically. Approximate the result to three decimal places.\(6^{5 x}=3000\)

Evaluate the logarithm. Round your result to three decimal places.\(\log _{20} 1575\)

Evaluate the expression without using a calculator.\(\log _{6} \frac{1}{36}\)

Sketch the graph of the function.\(y=e^{0.2 x}\)

Radioactive Decay What percent of a present amount of radioactive cesium \(\left({ }^{137} \mathrm{Cs}\right)\) will remain after 100 years? Use the fact that radioactive cesium has a half-life of 30 years.

Solve the exponential equation algebraically. Approximate the result to three decimal places.\(5^{-t / 2}=0.20\)

Evaluate the logarithm. Round your result to three decimal places.\(\log _{5} \frac{1}{3}\)

Evaluate the expression without using a calculator.\(\log _{8} 2\)

Sketch the graph of the function.\(f(x)=2 e^{0.12 x}\)

Solve the exponential equation algebraically. Approximate the result to three decimal places.\(4^{-t / 3}=0.15\)

Evaluate the logarithm. Round your result to three decimal places.\(\log _{9} \frac{3}{5}\)

Evaluate the expression without using a calculator.\(\log _{64} 4\)

Sketch the graph of the function.\(f(x)=3 e^{-0.2 x}\)

Find the constants \(C\) and \(k\) such that the exponential function \(y=C e^{k t}\) passes through the points on the graph.Learning Curve The management at a factory has found that the maximum number of units a worker can produce in a day is 40 . The learning curve for the number of units \(N\) produced per day after a new employee has worked \(t\) days is given by \(N=40\left(1-e^{k t}\right)\) After 20 days on the job, a particular worker produced 25 units in 1 day. (a) Find the learning curve for this worker (first find the value of \(k\) ). (b) How many days should pass before this worker is producing 35 units per day?

Solve the exponential equation algebraically. Approximate the result to three decimal places.\(3^{x-1}=28\)

Evaluate the logarithm. Round your result to three decimal places.\(\log _{1 / 4} 10\)

Evaluate the expression without using a calculator.\(\log _{7} 7\)

Sketch the graph of the function.\(f(x)=e^{2 x}\)

Solve the exponential equation algebraically. Approximate the result to three decimal places.\(2^{x-3}=31\)

Evaluate the logarithm. Round your result to three decimal places.\(\log _{1 / 3} 5\)

Evaluate the expression without using a calculator.\(\log _{12} 1\)

Sketch the graph of the function.\(h(x)=e^{x-2}\)

Motorola The sales per share \(S\) (in dollars) for Motorola from 1992 to 2005 can be approximated by the function \(S=\left\\{\begin{array}{lr}2.33-0.909 t+10.394 \ln t, & 2 \leq t \leq 10 \\ 0.6157 t^{2}-15.597 t+110.25, & 11 \leq t \leq 15\end{array}\right.\) where \(t\) represents the year, with \(t=2\) corresponding to 1992\. (Source: Motorola) (a) Use a graphing utility to graph the function. (b) Describe the change in sales per share that occurred in 2001 .

Solve the exponential equation algebraically. Approximate the result to three decimal places.\(2^{3-x}=565\)

Evaluate the logarithm. Round your result to three decimal places.\(\log _{1 / 2} 0.2\)

Evaluate the expression without using a calculator.\(\log _{10} 0.0001\)

Sketch the graph of the function.\(g(x)=1+e^{-x}\)

Intel The sales per share \(S\) (in dollars) for Intel from 1992 to 2005 can be approximated by the function \(S=\left\\{\begin{array}{lr}-1.48+2.65 \ln t, & 2 \leq t \leq 10 \\ 0.1586 t^{2}-3.465 t+22.87, & 11 \leq t \leq 15\end{array}\right.\) where \(t\) represents the year, with \(t=2\) corresponding to 1992\. (Source: Intel)

Evaluate the logarithm. Round your result to three decimal places.\(\log _{1 / 6} 0.025\)

Evaluate the expression without using a calculator.\(\log _{10} 100\)

Sketch the graph of the function.\(N(t)=1000 e^{-0.2 t}\)

Women's Heights The distribution of heights of American women (between 30 and 39 years of age) can be approximated by the function \(p=0.163 e^{-(x-64.9)^{2} / 12.03}, \quad 60 \leq x \leq 74\) where \(x\) is the height (in inches) and \(p\) is the percent (in decimal form). Use a graphing utility to graph the function. Then determine the average height of women in this age bracket. (Source: U.S. National Center for Health Statistics)

Solve the exponential equation algebraically. Approximate the result to three decimal places.\(8\left(10^{3 x}\right)=12\)

Approximate the logarithm using the properties of logarithms, given \(\log _{b} 2 \approx 0.3562\), \(\log _{b} 3 \approx 0.5646\), and \(\log _{b} 5 \approx 0.8271$$\log _{b} 10\)

Evaluate the expression without using a calculator.\(\ln e\)

Men's Heights The distribution of heights of American men (between 30 and 39 years of age) can be approximated by the function \(p=0.131 e^{-(x-69.9)^{2} / 18.66}, \quad 63 \leq x \leq 77\) where \(x\) is the height (in inches) and \(p\) is the percent (in decimal form). Use a graphing utility to graph the function. Then determine the average height of men in this age bracket. (Source: U.S. National Center for Health Statistics)

Solve the exponential equation algebraically. Approximate the result to three decimal places.\(5\left(10^{x-6}\right)=7\)

Approximate the logarithm using the properties of logarithms, given \(\log _{b} 2 \approx 0.3562\), \(\log _{b} 3 \approx 0.5646\), and \(\log _{b} 5 \approx 0.8271\)

Evaluate the expression without using a calculator.\(\ln e^{10}\)

Stocking a Lake with Fish \(\quad\) A lake is stocked with 500 fish, and the fish population \(P\) increases according to the logistic curve \(P=\frac{10,000}{1+19 e^{-t / 5}}, \quad t \geq 0\) where \(t\) is the time (in months).

Solve the exponential equation algebraically. Approximate the result to three decimal places.\(3\left(5^{x-1}\right)=21\)

Approximate the logarithm using the properties of logarithms, given \(\log _{b} 2 \approx 0.3562\), \(\log _{b} 3 \approx 0.5646\), and \(\log _{b} 5 \approx 0.8271$$\log _{b} \frac{2}{3}\)