Logarithmic Properties and Equations

Three properties, one origin

Each log property is an exponent rule read through the inverse:

• Product: $\log_b(xy)=\log_b x+\log_b y$ — because $b^m\cdot b^n=b^{m+n}$.
• Quotient: $\log_b\frac{x}{y}=\log_b x-\log_b y$.
• Power: $\log_b\left(x^n\right)=n\log_b x$.

plus the change of base $\log_b x=\dfrac{\log x}{\log b}$, which is how a calculator evaluates any base.

Solving equations

Two standard shapes. Exponential ($b^{f(x)}=c^{g(x)}$): take a log of both sides, pull exponents down with the power property, solve the resulting linear (or quadratic) equation. Logarithmic: condense to a single log on each side, translate to exponential form, solve — then check every candidate against the original domain. Squaring never appears, yet extraneous solutions do: condensing $\log(x-1)+\log(x-3)$ into $\log\left[(x-1)(x-3)\right]$ silently enlarges the domain, because the product can be positive while each factor is negative. Candidates that violate the original equation's domain must be discarded explicitly.