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Solving logarithmic equations

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Solving Logarithmic Equations

A logarithmic equation has a variable inside a logarithm. Solve it by condensing to a single logarithm with the log rules, rewriting log base b of x equals y as b to the y equals x, solving the result, and checking that every argument is positive. Includes a worked example.

What a logarithmic equation is

A logarithmic equation is an equation that has a variable inside a logarithm, such as log₂(x) = 3. To solve one you undo the logarithm and find the value of the variable. It helps to be comfortable with what a logarithm is before you start.

Most logarithmic equations follow the same four steps.

Steps to solve a logarithmic equation Step 1: condense to a single logarithm using log rules. Step 2: rewrite log base b of x equals y as b to the power y equals x. Step 3: solve the resulting equation. Step 4: check that the argument of every logarithm is positive. Example: log base 2 of x equals 3 becomes 2 cubed equals x, so x equals 8. 1Condense to a single logarithm (use log rules) 2Rewrite logb(x) = yasby= x 3Solve the resulting equation 4Check: the argument of every log must be > 0 Example log2(x) = 3 → 23= x → x = 8
Solving a logarithmic equation: condense, rewrite in exponential form, solve, then check.

The key move is step 2: converting from logarithmic to exponential form. Because logb(x) = y means exactly the same thing as by = x, rewriting the equation this way removes the logarithm and leaves an ordinary equation to solve.

When an equation has more than one logarithm on a side, first combine them into one using the laws of logarithms — for example the product rule of logarithms, which turns log(a) + log(b) into log(ab). Once each side is a single logarithm (or a logarithm equal to a number), you can rewrite in exponential form.

Every logarithm requires a positive argument, so always substitute your answer back in. Any solution that makes the inside of a logarithm zero or negative is extraneous and must be rejected.

Solve log₂(x) = 3. Rewrite in exponential form: 2³ = x, so x = 8. Check: the argument 8 is positive, so x = 8 is valid.

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