Logarithms - Examples & Applications

This section provides step-by-step examples of working with logarithms. You will learn 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.

Definition of a Logarithm

A logarithm is fundamentally an exponent. The equation y=log⁡b(x)y = \log_b(x) means exactly the same thing as by=xb^y = x. Converting between these two forms is the key to evaluating basic logs.

Example

Example 1: Converting Forms (Basic) Convert 25=322^5 = 32 to logarithmic form and log⁡3(81)=4\log_3(81) = 4 to exponential form.

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Example

Example 2: Evaluating Basic Logarithms (Intermediate) Evaluate the following without a calculator:
  1. log⁡4(64)\log_4(64)
  2. log⁡5(125)\log_5\left(\frac{1}{25}\right)
  3. log⁡16(4)\log_{16}(4)

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Example

Example 3: Logarithms with Radical Arguments (Advanced) Evaluate: log⁡2(163)\log_2(\sqrt[3]{16})

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Properties of Logarithms

The properties of logarithms (Product, Quotient, and Power Rules) are used to expand single logs into longer expressions or condense long expressions into a single logarithm.

Example

Example 1: Expanding a Logarithmic Expression (Intermediate) Use the properties of logarithms to expand the expression completely: log⁡b(x3y2z)\log_b\left(\frac{x^3y^2}{z}\right)

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Example

Example 2: Condensing a Logarithmic Expression (Intermediate) Write the expression as a single logarithm: 2ln⁡(x)−12ln⁡(y)+3ln⁡(z)2\ln(x) - \frac{1}{2}\ln(y) + 3\ln(z)

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Example

Example 3: Change of Base Formula (Advanced) Evaluate log⁡7(50)\log_7(50) using common logarithms (base 10) to four decimal places.

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Solving Exponential Equations

Exponential equations often require taking the logarithm of both sides to isolate the variable.

Example

Example 1: Using Logarithms to Solve (Advanced) Solve for xx: 5x+2=72x5^{x+2} = 7^{2x}

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Solving Logarithmic Equations

Solving equations involving logarithms requires condensing multiple logs into one, isolating the log term, and then converting the equation to exponential form to solve for the variable.

Checking Extraneous Solutions

The domain of a logarithm log⁡b(x)\log_b(x) requires that the argument xx must be strictly greater than zero (x>0x > 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.

Example

Example 1: Solving a Logarithmic Equation (Advanced) Solve for xx: log⁡2(x)+log⁡2(x−3)=2\log_2(x) + \log_2(x - 3) = 2

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Logarithmic Scale

Logarithmic scales compress wide ranges of values into a manageable linear scale, like the Richter scale.

Example

Example 1: The Richter Scale (Advanced) If an earthquake measures 6.06.0 on the Richter scale (R=log⁡(I/I0)R = \log(I / I_0)), how many times more intense is it than the reference earthquake?

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