Exponential and Log Equations

Key idea

Take logs or exponentiate both sides.

In depth

If the unknown is in an exponent, take logs of both sides. If it is inside a log, rewrite in exponential form. Check that arguments of logs are positive.

Worked example

3ˣ = 20 → x = log 20/log 3 ≈ 2.727. And log₂(x) = 5 → x = 32.

Real-world uses

• Finding how long an investment takes to double
• Calculating half-life of a substance

Common mistake

Accepting a solution that makes a log argument negative.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related advanced topics in Algebra

• Absolute Value Inequalities – Rewrite as a compound inequality.
• Systems with Three Variables – Reduce to two variables then solve in sequence.
• Systems of Inequalities – Solution region is the overlap of half-planes.
• Factoring by Grouping – Group terms in pairs to find a common binomial.
• Sum and Difference of Cubes – Special factoring patterns for cubes.
• Completing the Square – Rewrite a quadratic as a perfect square plus constant.