Logarithmic Equations: Solve Them Step by Step Easily

A logarithmic equation contains the unknown inside a logarithm, like log x + log(x − 2) = log 8. The standard strategy is to use the laws of logarithms to merge everything into a single logarithm, then drop the logs from both sides.

They matter because exponential relationships — population growth, radioactive decay, sound intensity — turn into logarithmic equations that are far easier to solve than the exponential forms they came from.

Key Concepts

• Domain first: the argument of a logarithm must be strictly positive, so log x demands x > 0 before any solving begins.
• Base conditions: the base must be positive and different from 1; school problems almost always use base 10.
• Merging with log laws: sums become products inside one log, differences become quotients.
• Dropping the logs: if log_a M = log_a N with a valid common base, then M = N.
• Extraneous solutions: values that satisfy the merged equation but violate the original domain must be rejected.

Formulas

These are the tools for merging and unmerging. Every formula needs a valid base a (a > 0, a ≠ 1) and positive arguments; applying them outside the domain is the main source of wrong answers.

• log_a M + log_a N = log_a(MN)
• log_a M − log_a N = log_a(M/N)
• n · log_a M = log_a(Mⁿ)
• If log_a M = log_a N then M = N

Worked Example

Solve log(x) + log(x − 2) = log 8 (base 10). First note the domain: x > 0 and x − 2 > 0, so x > 2. Merge the left side with the product rule: log[x(x − 2)] = log 8. Drop the logs: x(x − 2) = 8, giving x² − 2x − 8 = 0. Factorise: (x − 4)(x + 2) = 0, so x = 4 or x = −2. Domain check: x = −2 fails x > 2 (log(−2) is undefined), so it is extraneous and rejected. Only x = 4 survives. Verify: log 4 + log 2 = log(4 × 2) = log 8, correct.

Common Misconceptions

• Merging logs with different bases, e.g., log_2 x + log_3 x — the laws require one common base on every term.
• Keeping x = −2 in the example. Every candidate must be checked against the original equation's domain, not just the merged form.
• Writing log(x²) = 2 log x for negative x. The power law needs x > 0; for x < 0 the correct form is log(x²) = 2 log|x|.

Real-World Applications

• Solving exponential growth and decay models for time or rate.
• pH calculations: recovering hydrogen-ion concentration from pH.
• Decibel and Richter scale problems converting levels back to intensities.
• Half-life computations in nuclear physics and medicine.

To solve a logarithmic equation, fix the domain first, merge with the log laws, drop the logs, solve the resulting equation, and reject anything outside the domain. The domain check is not optional — it is where extraneous solutions are caught.