Solving Exponential Equations with Compatible Bases

Objective: Solve by equating exponents for an injective exponential base.

Prerequisite skills: exponent laws and solving linear equations. For b>0 and b≠1, the function b^x takes each positive output only once. Therefore b^u=b^v implies u=v. This avoids logarithms when both sides have compatible bases. Common error: equating exponents while the bases still differ. Restrictions and interpretation: Match powers only after obtaining equal positive bases other than 1.

Objective: Express a power equation using one common base.

Write bases such as 4, 8, and 16 as powers of 2 before applying the equal-base principle. A power raised to another exponent multiplies the exponents. Common error: equating exponents of 4 and 8 without converting bases. Restrictions and interpretation: Match powers only after obtaining equal positive bases other than 1.

Objective: Solve a linear exponent equation after rewriting both sides.

After rewriting in a common base, exponent expressions may contain x on both sides. Solve that linear equation carefully, then verify the resulting exponent values agree. No approximate graph reading is necessary. Common error: dropping an exponent offset during the common-base conversion. Restrictions and interpretation: Match powers only after obtaining equal positive bases other than 1. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.