Chapter 4

Write the logarithm in terms of natural logarithms.\(\log _{10} 20\)

Use the definition of a logarithm to write the equation in logarithmic form. For example, the logarithmic form of \(2^{3}=8\) is \(\log _{2} 8=3$$e^{-x}=2\)

Classify the model as an exponential growth model or an exponential decay model.\(y=3 e^{0.5 t}\)

Apply the Inverse Property of logarithmic or exponential functions to simplify the expression.\(-8+e^{\ln x^{3}}\)

Write the logarithm in terms of natural logarithms.\(\log _{3} n\)

Use the definition of a logarithm to write the equation in exponential form. For example, the exponential form of \(\log _{5} 125=3\) is \(5^{3}=125\).\(\log _{4} 16=2\)

Classify the model as an exponential growth model or an exponential decay model.\(y=2 e^{-0.6 t}\)

Apply the Inverse Property of logarithmic or exponential functions to simplify the expression.\(-1+\ln e^{2 x}\)

Write the logarithm in terms of natural logarithms.\(\log _{2} m\)

Use the definition of a logarithm to write the equation in exponential form. For example, the exponential form of \(\log _{5} 125=3\) is \(5^{3}=125\).\(\log _{10} 1000=3\)

Classify the model as an exponential growth model or an exponential decay model.\(y=20 e^{-1.5 t}\)

Apply the Inverse Property of logarithmic or exponential functions to simplify the expression.\(10^{\log _{10}(x+5)}\)

Write the logarithm in terms of natural logarithms.\(\log _{1 / 5} x\)

Use the definition of a logarithm to write the equation in exponential form. For example, the exponential form of \(\log _{5} 125=3\) is \(5^{3}=125\).\(\log _{2} \frac{1}{2}=-1\)

Sketch the graph of the function.\(g(x)=4^{x}\)

Classify the model as an exponential growth model or an exponential decay model.\(y=4 e^{0.07 t}\)

Apply the Inverse Property of logarithmic or exponential functions to simplify the expression.\(10^{\log _{10}\left(x^{2}+7 x+10\right)}\)

Write the logarithm in terms of natural logarithms.\(\log _{1 / 3} x\)

Use the definition of a logarithm to write the equation in exponential form. For example, the exponential form of \(\log _{5} 125=3\) is \(5^{3}=125\).\(\log _{3} \frac{1}{9}=-2\)

Sketch the graph of the function.\(f(x)=\left(\frac{3}{2}\right)^{x}\)

Apply the Inverse Property of logarithmic or exponential functions to simplify the expression.\(2^{\log _{2} x^{2}}\)

Use the definition of a logarithm to write the equation in exponential form. For example, the exponential form of \(\log _{5} 125=3\) is \(5^{3}=125\).\(\ln e=1\)

Sketch the graph of the function.\(f(x)=4^{-x}\)

Apply the Inverse Property of logarithmic or exponential functions to simplify the expression.\(9^{\log _{9}(3 x+7)}\)

Write the logarithm in terms of natural logarithms.\(\log _{x} \frac{3}{4}\)

Use the definition of a logarithm to write the equation in exponential form. For example, the exponential form of \(\log _{5} 125=3\) is \(5^{3}=125\).\(\ln \frac{1}{e}=-1\)

Sketch the graph of the function.\(h(x)=\left(\frac{3}{2}\right)^{-x}\)

Solve the exponential equation algebraically. Approximate the result to three decimal places.\(3 e^{x}=9\)

Write the logarithm in terms of natural logarithms.\(\log _{2.6} x\)

Use the definition of a logarithm to write the equation in exponential form. For example, the exponential form of \(\log _{5} 125=3\) is \(5^{3}=125\).\(\log _{5} 0.2=-1\)

Sketch the graph of the function.\(h(x)=4^{x-3}\)

Solve the exponential equation algebraically. Approximate the result to three decimal places.\(5 e^{x}=20\)

Write the logarithm in terms of natural logarithms.\(\log _{7.1} x\)

Use the definition of a logarithm to write the equation in exponential form. For example, the exponential form of \(\log _{5} 125=3\) is \(5^{3}=125\).\(\log _{10} 0.1=-1\)

Sketch the graph of the function.\(g(x)=\left(\frac{3}{2}\right)^{x+2}\)

Population The population \(P\) of a city is given by \(P=120,000 e^{0.016 t}\) where \(t\) represents the year, with \(t=0\) corresponding to 2000\. Sketch the graph of this equation. Use the model to predict the year in which the population of the city will reach about 180,000

Solve the exponential equation algebraically. Approximate the result to three decimal places.\(2\left(3^{x}\right)=16\)

Evaluate the logarithm. Round your result to three decimal places.\(\log _{2} 6\)

Use the definition of a logarithm to write the equation in exponential form. For example, the exponential form of \(\log _{5} 125=3\) is \(5^{3}=125\).\(\log _{27} 3=\frac{1}{3}\)

Sketch the graph of the function.\(g(x)=4^{-x}-2\)

Solve the exponential equation algebraically. Approximate the result to three decimal places.\(3\left(4^{x}\right)=81\)

Evaluate the logarithm. Round your result to three decimal places.\(\log _{8} 3\)

Use the definition of a logarithm to write the equation in exponential form. For example, the exponential form of \(\log _{5} 125=3\) is \(5^{3}=125\).\(\log _{8} 2=\frac{1}{3}\)

Sketch the graph of the function.\(f(x)=\left(\frac{3}{2}\right)^{-x}+2\)

Bacteria Growth The number \(N\) of bacteria in a culture is given by the model \(N=100 e^{k t}\), where \(t\) is the time (in hours), with \(t=0\) corresponding to the time when \(N=100\). When \(t=6\), there are 140 bacteria. How long does it take the bacteria population to double in size? To triple in size?

Solve the exponential equation algebraically. Approximate the result to three decimal places.\(e^{x}-9=19\)

Evaluate the logarithm. Round your result to three decimal places.\(\log _{27} 35\)

Evaluate the expression without using a calculator.\(\log _{3} 9\)

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

Bacteria Growth The number \(N\) of bacteria in a culture is given by the model \(N=250 e^{k t}\), where \(t\) is the time (in hours), with \(t=0\) corresponding to the time when \(N=250\). When \(t=10\), there are 320 bacteria. How long does it take the bacteria population to double in size? To triple in size?