Logarithmic Equations

Solve the algebra, then enforce the logarithm's domain: every argument must remain positive.

Domain conditions first

For real logarithms, each argument must satisfy

Record these restrictions before combining or exponentiating. Algebraic manipulation can produce candidates that make an original logarithm undefined.

Convert one logarithm to exponential form

If

then

For example,

becomes $x-1=9$, so $x=10$. Since $x-1>0$, the solution is valid.

Use the one-to-one property

If two logs with the same valid base are equal,

then $A=B$, provided $A>0$ and $B>0$.

Thus $\log_5(2x-1)=\log_5(x+7)$ gives $2x-1=x+7$, so $x=8$, which satisfies both argument conditions.

Combining logarithms

Use

and

to create one logarithm. A coefficient becomes an exponent through $p\log_bM=\log_b(M^p)$.

These laws apply only when all original arguments are positive.

Worked example with an extraneous root

The rejected value solves the transformed quadratic but not the original logarithmic equation.

Equations involving different bases

If simple rewriting gives a common base, use it. Otherwise apply change of base or exponentiate after isolating a logarithm.

For numerical work,

Keep full calculator precision until the final line and substitute the approximation back into the original equation.

Mixed logarithmic and algebraic terms

An equation such as

may not yield to elementary algebra. Graph $y=\log_2x$ and $y=x-2$ or use numerical methods, while keeping $x>0$. Multiple intersections may exist, so choose a window and method that can find all solutions in the requested interval.

Exponential equations via logarithms

Logarithms also solve equations with variables in exponents:

Taking logs gives

The same inverse relationship works in both directions.

Applications

Logarithmic equations appear when solving for time in compound growth or decay, interpreting pH and sound levels, and inverting exponential models. State the units and distinguish a continuous time from the first completed discrete period.

Real-world scales may have definitions containing extra coefficients, so use the given model rather than assuming every scale is simply $\log x$.

Common mistakes

Ignoring argument restrictions. Check every original logarithm.

Splitting $\log(M+N)$. No sum law exists.

Dropping coefficients instead of turning them into powers. Use the power law.

Keeping every polynomial root. Transformed equations can introduce invalid candidates.

Rounding before verification. Preserve precision, then check the original.

Quick self-check

• What inequalities keep all original arguments positive?
• Can the logs be combined legally?
• Should I convert to exponential form or use one-to-one reasoning?
• Have all algebraic candidates been tested against the domain?
• If the answer is approximate, does it verify to the requested precision?

Related topics

• Logarithmic Functions
• Exponential Equations
• Inverse Functions
• Exponents