Chapter 8

Find \(x\) to the nearest hundredth. \(2 \log x=\log (x-1)+\log 5\)

In \(15-23,\) evaluate each logarithm to the nearest hundredth. $$ \log 0.002 $$

In \(15-26,\) write each logarithmic equation in exponential form. $$ \log _{4} 16=2 $$

a. Write each expression as a single logarithm. b. Find the value of each expression. \(\log _{3} 243-\log _{3} 729\)

In \(11-22,\) solve each equation for \(y\) in terms of \(x\) $$ x=12^{-y} $$

In \(15-20,\) evaluate each logarithm to the nearest hundredth. $$ \ln 1,000^{2} $$

After how many years will \(\$ 100\) invested at an annual rate of 6\(\%\) compounded continuously be worth at least \(\$ 450 ?\) (Use the formula \(A_{n}=A_{0} e^{t r} . )\)

Find \(x\) to the nearest hundredth. \(2 \log x=\log (x+3)+\log 2\)

In \(15-23,\) evaluate each logarithm to the nearest hundredth. $$ \log 9+\log 3 $$

In \(15-26,\) write each logarithmic equation in exponential form. $$ 7=\log _{2} 128 $$

a. Write each expression as a single logarithm. b. Find the value of each expression. \(\log _{3} 6,561-\log _{3} 243\)

In \(11-22,\) solve each equation for \(y\) in terms of \(x\) $$ x=\log _{2} y $$

In \(15-20,\) evaluate each logarithm to the nearest hundredth. $$ \frac{\ln 6-\ln e}{2 \ln 8} $$

The decay constant of francium is \(-0.0315\) minutes. a. After how many minutes will 1.25 grams of francium remain of a 10.0 -gram sample? Assume the exponential decay occurs continuously. b. What is the half-life of francium? (The half-life of an element is the length of time needed for half of a sample to decay. For example, it is the length of time for a sample of 10 grams to be reduced to 5 grams of the original element.)

In \(15-23,\) evaluate each logarithm to the nearest hundredth. $$ \log 64-\log 16 $$

In \(15-26,\) write each logarithmic equation in exponential form. $$ 5=\log _{3} 243 $$

a. Write each expression as a single logarithm. b. Find the value of each expression. \(\frac{1}{3} \log _{3} 2,187+\frac{1}{6} \log _{3} 81\)

In \(11-22,\) solve each equation for \(y\) in terms of \(x\) $$ x=\log _{5} y $$

In \(15-20,\) evaluate each logarithm to the nearest hundredth. $$ \frac{\ln \sqrt{5}}{\ln 10} $$

The half-life of einsteinium is 276 days. a. To five decimal places, what is the decay constant of einsteinium? Assume the exponential decay occurs continuously. b. After how many days will 2.5 grams of einsteinium remain of a sample of 20 grams?

In \(15-23,\) evaluate each logarithm to the nearest hundredth. $$ 200 \log \frac{5}{2} $$

In \(15-26,\) write each logarithmic equation in exponential form. $$ \log _{7} 1=0 $$

a. Write each expression as a single logarithm. b. Find the value of each expression. \(\log _{3} 9-2 \log _{3} 27+\log _{3} 243\)

In \(11-22,\) solve each equation for \(y\) in terms of \(x\) $$ x=\log _{10} y $$

In \(21-32,\) for each given logarithm, find \(x,\) the antilogarithm. Write the answer to four decimal places. $$ \ln x=0.5787 $$

In \(15-23,\) evaluate each logarithm to the nearest hundredth. $$ \frac{3 \log 4}{\log 5} $$

In \(15-26,\) write each logarithmic equation in exponential form. $$ \log _{10} 0.001=-3 $$

a. Expand each expression as a difference, sum, and/or multiple of logarithms. b. Find the value of each expression. 4 \(\log _{3} \frac{9}{27}\)

In \(11-22,\) solve each equation for \(y\) in terms of \(x\) $$ x=\log _{8} y $$

In \(21-32,\) for each given logarithm, find \(x,\) the antilogarithm. Write the answer to four decimal places. $$ \ln x=0.8297 $$

In \(15-23,\) evaluate each logarithm to the nearest hundredth. $$ \frac{\log 100-\frac{1}{2} \log 36}{\log 6} $$

In \(15-26,\) write each logarithmic equation in exponential form. $$ \log _{100} 0.01=-1 $$

a. Expand each expression as a difference, sum, and/or multiple of logarithms. b. Find the value of each expression. \(\frac{1}{2} \log _{3} 3(243)\)

In \(11-22,\) solve each equation for \(y\) in terms of \(x\) $$ x=\log _{0.1} y $$

In \(21-32,\) for each given logarithm, find \(x,\) the antilogarithm. Write the answer to four decimal places. $$ \ln x=1.3826 $$

In \(15-23,\) evaluate each logarithm to the nearest hundredth. $$ \log \frac{1}{2} \cdot \log 100 \cdot \log 300 $$

In \(15-26,\) write each logarithmic equation in exponential form. $$ -2=\log _{5} 0.04 $$

a. Expand each expression as a difference, sum, and/or multiple of logarithms. b. Find the value of each expression. \(\log _{4} \sqrt{16^{2}}\)

In \(21-32,\) for each given logarithm, find \(x,\) the antilogarithm. Write the answer to four decimal places. $$ \ln x=1.7790 $$

In \(24-35,\) for each given logarithm, find the antilogarithm, \(x .\) Write the answer to four decimal places. $$ \log x=0.5787 $$

In \(15-26,\) write each logarithmic equation in exponential form. $$ \log _{8} 2=\frac{1}{3} $$

Write each expression as a single logarithm. \(\log _{e} x+\log _{e} 10\)

If money is invested at a rate of 5\(\%\) compounded annually, then for each dollar invested, the amount of money in an account is \(g(x),\) when \(g(x)=1.05^{x}\) after \(x\) years. a. Write the ordered pairs of the function \(\mathrm{g}\) for \(0 \leq x \leq 3\) and locate the pairs as points on a graph. The domain is the set of non- negative integers. b. Write the ordered pairs for \(\mathrm{g}^{-1}(x)\) and sketch the graph.

In \(21-32,\) for each given logarithm, find \(x,\) the antilogarithm. Write the answer to four decimal places. $$ \ln x=2.2030 $$

In \(24-35,\) for each given logarithm, find the antilogarithm, \(x .\) Write the answer to four decimal places. $$ \log x=0.8297 $$

In \(15-26,\) write each logarithmic equation in exponential form. $$ \log _{49} 343=\frac{3}{2} $$

Write each expression as a single logarithm. \(\log _{2} a+\log _{2} b\)

In \(21-32,\) for each given logarithm, find \(x,\) the antilogarithm. Write the answer to four decimal places. $$ \ln x=2.5619 $$

In \(24-35,\) for each given logarithm, find the antilogarithm, \(x .\) Write the answer to four decimal places. $$ \log x=1.3826 $$

In \(15-26,\) write each logarithmic equation in exponential form. $$ -\frac{2}{5}=\log _{32} 0.25 $$