6.3 Logarithmic Functions

Find the inverse of $f \left(x\right) = 2^{x}$

Convert between exponential and logarithmic form.

ⓐ Convert to logarithmic form: $2^{3} = 8$

Convert between exponential and logarithmic form.

ⓑ Convert to exponential form: $\text{log}_{b} m = a$

Find the value of x: ⓐ $\text{log}_{x} 36 = 2 ,$ ⓑ $\text{log}_{4} x = 3 ,$ and ⓒ $\text{log}_{\frac{1}{2}} \frac{1}{8} = x .$

Write the following logarithmic equations in exponential form.

1. ⓐ $\log_{6} \left(\sqrt{6}\right) = \frac{1}{2}$
2. ⓑ $\log_{3} \left(9\right) = 2$

Write the following logarithmic equations in exponential form.

1. ⓐ$\log_{10} \left(1, 000, 000\right) = 6$
2. ⓑ$\log_{5} \left(25\right) = 2$

Write the following exponential equations in logarithmic form.

1. $2^{3} = 8$
2. $5^{2} = 25$
3. $10^{- 4} = \frac{1}{10,000}$

Write the following exponential equations in logarithmic form.

1. ⓐ $3^{2} = 9$
2. ⓑ $5^{3} = 125$
3. ⓒ $2^{- 1} = \frac{1}{2}$

Solve $y = \log_{4} \left(64\right)$ without using a calculator.

Solve $y = \log_{121} \left(11\right)$ without using a calculator.

Evaluate $y = \log_{3} \left(\frac{1}{27}\right)$ without using a calculator.

Evaluate $y = \log_{2} \left(\frac{1}{32}\right)$ without using a calculator.

Evaluate $y = \log \left(1000\right)$ without using a calculator.

Evaluate $y = \log \left(1, 000, 000\right) .$

Evaluate $y = \log \left(321\right)$ to four decimal places using a calculator.

Evaluate $y = \log \left(123\right)$ to four decimal places using a calculator.

The amount of energy released from one earthquake was 500 times greater than the amount of energy released from another. The equation $10^{x} = 500$ represents this situation, where $x$ is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?

The amount of energy released from one earthquake was $\text{8},\text{500}$ times greater than the amount of energy released from another. The equation $10^{x} = 8500$ represents this situation, where $x$ is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?

Evaluate $y = \ln \left(500\right)$ to four decimal places using a calculator.

Evaluate $\ln \left(−500\right) .$

What is a base $b$ logarithm? Discuss the meaning by interpreting each part of the equivalent equations $b^{y} = x$ and $\log_{b} x = y$ for $b > 0 , b \neq 1.$

How can the logarithmic equation $\log_{b} x = y$ be solved for $x$ using the properties of exponents?

Discuss the meaning of the natural logarithm. What is its relationship to a logarithm with base $b ,$ and how does the notation differ?

Rewrite each equation in exponential form.

$\text{log}_{a} \left(b\right) = c$

Rewrite each equation in exponential form.

$\log_{x} \left(64\right) = y$

Rewrite each equation in exponential form.

$\log_{15} \left(a\right) = b$

Rewrite each equation in exponential form.

$\log_{13} \left(142\right) = a$

Rewrite each equation in exponential form.

$\text{ln} \left(w\right) = n$

Rewrite each equation in logarithmic form.

$c^{d} = k$

Rewrite each equation in logarithmic form.

$19^{x} = y$

Rewrite each equation in logarithmic form.

$n^{4} = 103$

Rewrite each equation in logarithmic form.

$y^{x} = \frac{39}{100}$

Rewrite each equation in logarithmic form.

$e^{k} = h$

Solve for $x$ by converting the logarithmic equation to exponential form.

$\text{log}_{2} \left(x\right) = - 3$

Solve for $x$ by converting the logarithmic equation to exponential form.

$\log_{3} \left(x\right) = 3$

Solve for $x$ by converting the logarithmic equation to exponential form.

$\text{log}_{9} \left(x\right) = \frac{1}{2}$

Solve for $x$ by converting the logarithmic equation to exponential form.

$\log_{6} \left(x\right) = - 3$

Solve for $x$ by converting the logarithmic equation to exponential form.

$\text{ln} \left(x\right) = 2$

Use the definition of common and natural logarithms to simplify.

$10^{\text{log} \left(32\right)}$

Use the definition of common and natural logarithms to simplify.

$e^{\ln \left(1.06\right)}$

Use the definition of common and natural logarithms to simplify.

$e^{\ln \left(10.125\right)} + 4$

Evaluate the base $b$ logarithmic expression without using a calculator.

$\text{log}_{6} \left(\sqrt{6}\right)$

Evaluate the base $b$ logarithmic expression without using a calculator.

$6 \text{log}_{8} \left(4\right)$

Evaluate the common logarithmic expression without using a calculator.

$\text{log} \left(0.001\right)$

Evaluate the common logarithmic expression without using a calculator.

$2 \text{log} \left(100^{- 3}\right)$

Evaluate the natural logarithmic expression without using a calculator.

$\text{ln} \left(1\right)$

Evaluate the natural logarithmic expression without using a calculator.

$25 \text{ln} \left(e^{\frac{2}{5}}\right)$

Evaluate each expression using a calculator. Round to the nearest thousandth.

$\text{ln} \left(15\right)$

Evaluate each expression using a calculator. Round to the nearest thousandth.

$\text{log} \left(\sqrt{2}\right)$

Is $x = 0$ in the domain of the function $f \left(x\right) = \log \left(x\right) ?$ If so, what is the value of the function when $x = 0 ?$ Verify the result.

Is there a number $x$ such that $\ln x = 2 ?$ If so, what is that number? Verify the result.

Is the following true: $\frac{\ln \left(e^{1.725}\right)}{\ln \left(1\right)} = 1.725 ?$ Verify the result.

Refer to the previous exercise. Suppose the light meter on a camera indicates an $E I$ of $- 2 ,$ and the desired exposure time is 16 seconds. What should the f-stop setting be?