Logarithms - Examples & Applications

This section provides step-by-step examples of working with logarithms, demonstrating how to convert between exponential and logarithmic forms, apply log properties to simplify complex expressions, and solve equations where the variable is trapped in an exponent or inside a log.

Example 1: Converting Forms (Basic)

Example

Problem: Convert $2^5 = 32$ to logarithmic form and $\log_3(81) = 4$ to exponential form.

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Example 2: Evaluating Basic Logarithms (Intermediate)

Example

Problem: Evaluate the following without a calculator:

$\log_4(64)$
$\log_5\left(\frac{1}{25}\right)$
$\log_{16}(4)$

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Example 3: Logarithms with Radical Arguments (Advanced)

Example

Problem: Evaluate $\log_2(\sqrt[3]{16})$ .

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Example 4: Expanding a Logarithmic Expression (Intermediate)

Example

Problem: Use the properties of logarithms to expand the expression completely: $\log_b\left(\frac{x^3y^2}{z}\right)$ .

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Example 5: Condensing a Logarithmic Expression (Intermediate)

Example

Problem: Write the expression as a single logarithm: $2\ln(x) - \frac{1}{2}\ln(y) + 3\ln(z)$ .

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Example 6: Change of Base Formula (Advanced)

Example

Problem: Evaluate $\log_7(50)$ using common logarithms (base $10$ ) to four decimal places.

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Example 7: Using Logarithms to Solve (Advanced)

Example

Problem: Solve for $x$ : $5^{x+2} = 7^{2x}$ .

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Example 8: Solving a Logarithmic Equation (Advanced)

Example

Problem: Solve for $x$ : $\log_2(x) + \log_2(x - 3) = 2$ .

Caution

The domain of a logarithm $\log_b(x)$ requires that the argument $x$ must be strictly greater than zero ( $x \gt 0$ ). When you solve a logarithmic equation, you MUST substitute your answers back into the original equation to ensure you aren't trying to take the log of a negative number or zero.

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Example 9: The Richter Scale (Advanced)

Example

Problem: If an earthquake measures $6.0$ on the Richter scale ( $R = \log_{10}(I / I_0)$ ), how many times more intense is it than the reference earthquake?

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