Solving Exponential Equations

Pre-Calculus · Axiom Academy

LESSON Solving Exponential Equations Two moves get the unknown out of the exponent: match the base, or take the log of both sides. 1. The Variable Is in the Exponent Compare two equations. In 2x = 8 the x is a factor, so one division frees it. In 2^x = 8 the x is the exponent — it controls how many times the base is multiplied, and no single arithmetic step pulls it out of that position. Linear: divide and you're done Exponential: x sits in the exponent Take 2^x = 8 . The right side is small enough to rewrite as a power of the same base : 8 = 2^3 . Once both sides read , the only way the two powers can be equal is for the exponents to be equal. Same base equate the exponents The exponent can be a whole expression. Solve 2^ x+1 = 8 . Same idea: rewrite 8 as a power of 2 , line up the bases, then set the exponents equal and finish with one-step algebra. Rewrite the right side as a power of 2: 8 = 2^3 . Now both sides have base 2, so the exponents must be equal. Solve the resulting linear equation. 4. Method 2 — Take the Log of Both Sides Now try 3^x = 7 . Is 7 a power of 3 ? No whole power works ( 3^1=3 , 3^2=9 ), so matching bases is hopeless. Instead, apply a logarithm to both sides . The power rule, , drops the exponent to the front, where ordinary algebra can reach it. Take the log (natural log here) of both sides. Power rule: the exponent x slides to the front. 5. Which Method Should You Use?

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