Business Mathematics and Basic Statistics · Ch 4 — Logarithms

Solving Simple Logarithmic and Exponential Equations

Converting between logarithmic and exponential form

The fastest way to solve a simple equation involving a logarithm is often to rewrite it in exponential form using the definition directly: $\log_a N=x \iff a^x=N$ . For example, $\log_3 x=4$ becomes $x=3^4=81$ immediately.

Using the laws to combine multiple logarithm terms first

When an equation has MORE than one logarithm term, use the product/quotient laws to combine them into a SINGLE logarithm before converting to exponential form. For example, to solve $\log x+\log(x-3)=1$ (base 10):

\log\big(x(x-3)\big) = 1 \ \Rightarrow\ x(x-3) = 10^1 = 10 \ \Rightarrow\ x^2-3x-10=0

Factorising: $(x-5)(x+2)=0$ , so $x=5$ or $x=-2$ .

Note

Always check a logarithmic equation's solutions against the ORIGINAL equation

A logarithm is only defined for a POSITIVE argument. In the example above, $x=-2$ would make $\log(x-3)=\log(-5)$ , the logarithm of a negative number — undefined — so $x=-2$ must be REJECTED even though it solves the quadratic correctly. Only $x=5$ (which gives $\log 5+\log 2$ , both defined) is a valid solution to the original logarithmic equation. This check is not optional — a quadratic derived from a log equation routinely produces one spurious root this way.

Solving a simple exponential equation using logs (when a common base isn't obvious) …

Definition 1Logarithmic Equation

An equation involving one or more logarithm terms. Solved by combining multiple log terms into one (product/quotient laws), converting to exponential form via the definition, then ALWAYS checking every solution against the domain of the original logarithm …

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