Condensing logs and changing base

Condensing logs and changing base

Combine logarithmic expressions into a single logarithm and evaluate unfamiliar bases without losing the original restrictions. This precalculus practice connects the product, quotient and power laws with the change-of-base formula.

Condense positive-variable expressions, fractional coefficients and ordinary constants. Preserve the domain when a new product formula appears to admit extra inputs, and distinguish expressions with absolute values from those requiring positive variables. Mixed-base terms must be converted before they are combined. Reciprocal and chained logarithmic factors provide exact checks on the change-of-base rule.

Prerequisites are logarithm definitions, valid bases and the expansion laws. Single-select and multiselect questions progress from algebraic condensation to domain decisions and exact identities. One numerical evaluation permits a scientific calculator and specifies the required rounding; the remaining work can be done exactly.

Write down the original argument conditions first, choose a common base when needed and round only the final numerical result. The worked explanations show why reversed fractions, mixed logarithm bases and lost restrictions can produce convincing but incorrect answers. This set prepares for exponential and logarithmic equation solving without requiring those later techniques.