Chapter 3

Solve the exponential equations exactly for \(x\). $$2^{x^{2}}=16$$

Apply the properties of logarithms to simplify each expression. Do not use a calculator. $$\log _{9} 1$$

Evaluate exactly (without using a calculator). For rational exponents, consider converting to radical form first. $$5^{-2}$$

Write each logarithmic equation in its equivalent exponential form. $$\log _{81} 3=\frac{1}{4}$$

Solve the exponential equations exactly for \(x\). $$169^{x}=13$$

Apply the properties of logarithms to simplify each expression. Do not use a calculator. $$\log _{69} 1$$

Evaluate exactly (without using a calculator). For rational exponents, consider converting to radical form first. $$4^{-3}$$

Write each logarithmic equation in its equivalent exponential form. $$\log _{121} 11=\frac{1}{2}$$

Solve the exponential equations exactly for \(x\). $$\left(\frac{2}{3}\right)^{x+1}=\frac{27}{8}$$

Apply the properties of logarithms to simplify each expression. Do not use a calculator. $$\log _{1 / 2}\left(\frac{1}{2}\right)$$

Evaluate exactly (without using a calculator). For rational exponents, consider converting to radical form first. $$8^{2 / 3}$$

Write each logarithmic equation in its equivalent exponential form. $$\log _{2}\left(\frac{1}{32}\right)=-5$$

Solve the exponential equations exactly for \(x\). $$\left(\frac{3}{5}\right)^{x+1}=\frac{25}{9}$$

Apply the properties of logarithms to simplify each expression. Do not use a calculator. $$\log _{3,3} 3.3$$

Evaluate exactly (without using a calculator). For rational exponents, consider converting to radical form first. $$27^{2 / 3}$$

Write each logarithmic equation in its equivalent exponential form. $$\log _{3}\left(\frac{1}{81}\right)=-4$$

Solve the exponential equations exactly for \(x\). $$e^{2 x+3}=1$$

Apply the properties of logarithms to simplify each expression. Do not use a calculator. $$\log _{10} 10^{8}$$

Evaluate exactly (without using a calculator). For rational exponents, consider converting to radical form first. $$\left(\frac{1}{9}\right)^{-3 / 2}$$

Write each logarithmic equation in its equivalent exponential form. $$\log 0.01=-2$$

Solve the exponential equations exactly for \(x\). $$10^{x^{2}-1}=1$$

Apply the properties of logarithms to simplify each expression. Do not use a calculator. $$\ln e^{3}$$

Evaluate exactly (without using a calculator). For rational exponents, consider converting to radical form first. $$\left(\frac{1}{16}\right)^{-3 / 2}$$

Write each logarithmic equation in its equivalent exponential form. $$\log 0.0001=-4$$

The population of the Philippines in 2003 was 80 million. It increases \(2.36 \%\) per year. What is the expected population of the Philippines in \(2010 ?\) Apply the formula \(N=N_{0} e^{r t},\) where \(N\) represents the number of people.

Solve the exponential equations exactly for \(x\). $$7^{2 x-5}=7^{3 x-4}$$

Apply the properties of logarithms to simplify each expression. Do not use a calculator. $$\log _{10} 0.001$$

Approximate with a calculator. Round your answer to four decimal places. $$5^{\sqrt{2}}$$

Write each logarithmic equation in its equivalent exponential form. $$\log 10,000=4$$

China's urban population is growing at 2.5\% a year, compounding continuously. If there were 13.7 million people in Shanghai in \(1996,\) approximately how many people will there be in \(2016 ?\) Apply the formula \(N=N_{0} e^{r t},\) where \(N\) represents the number of people.

Solve the exponential equations exactly for \(x\). $$125^{x}=5^{2 x-3}$$

Apply the properties of logarithms to simplify each expression. Do not use a calculator. $$\log _{3} 3^{7}$$

Approximate with a calculator. Round your answer to four decimal places. $$6^{\sqrt{3}}$$

Write each logarithmic equation in its equivalent exponential form. $$\log 1000=3$$

Solve the exponential equations exactly for \(x\). $$2^{x^{2}+12}=2^{7 x}$$

Apply the properties of logarithms to simplify each expression. Do not use a calculator. $$\log _{2} \sqrt{8}$$

Approximate with a calculator. Round your answer to four decimal places. $$e^{2}$$

Write each logarithmic equation in its equivalent exponential form. $$\log _{1 / 4}(64)=-3$$

Solve the exponential equations exactly for \(x\). $$5^{x^{2}-3}=5^{2 x}$$

Apply the properties of logarithms to simplify each expression. Do not use a calculator. $$\log _{5} \sqrt[3]{5}$$

Approximate with a calculator. Round your answer to four decimal places. $$e^{1 / 2}$$

Write each logarithmic equation in its equivalent exponential form. $$\log _{1 / 6}(36)=-2$$

The number of cell phones in China is exploding. In 2007 there were 487.4 million cell phone subscribers and the number is increasing at a rate of \(16.5 \%\) per year. How many cell phone subscribers are expected in \(2010 ?\) Use the formula \(N=N_{0} e^{r t},\) where \(N\) represents the number of cell phone subscribers. Let \(t=0\) correspond to 2007.

Solve the exponential equations exactly for \(x\). $$9^{x}=3^{x^{2}-4 x}$$

Apply the properties of logarithms to simplify each expression. Do not use a calculator. $$8^{\log _{8} 5}$$

Approximate with a calculator. Round your answer to four decimal places. $$e^{-\pi}$$

Write each logarithmic equation in its equivalent exponential form. $$-1=\ln \left(\frac{1}{e}\right)$$

A colony of bacteria is growing exponentially. Initially, 500 bacteria were in the colony. The growth rate is \(20 \%\) per hour. (a) How many bacteria should be in the colony in 12 hours? (b) How many in 1 day? Use the formula \(N=N_{0} e^{r t},\) where \(N\) represents the number of bacteria.

Solve the exponential equations exactly for \(x\). $$16^{x-1}=2^{x^{2}}$$

Apply the properties of logarithms to simplify each expression. Do not use a calculator. $$2^{\log _{2} 5}$$