Solving Exponential Equations Step by Step

Pre-Calculus · Axiom Academy

EXAMPLE Solving Exponential Equations When both sides can share a base, rewrite and equate the exponents Solve the exponential equation 2^ x+1 = 8^ x-2 for x . Both sides are powers of 2 , so we can match the bases instead of reaching for logarithms. Nice work. You matched bases on 2^ x+1 = 8^ x-2 and solved it without a single logarithm. Same base possible? Rewrite both sides with one common base, then set the exponents equal — here 8 = 2^3 turned the equation into x+1 = 3(x-2) . Different bases? Take the log of both sides and let the power rule bring the exponent down: . For example, 3^x = 15 gives . Isolate first. Strip away any coefficient before taking a log: becomes 2^x = 16 , so x = 4 . Result: 2^ x+1 = 8^ x-2 has the single solution x = 3.5 (both sides equal ). The variable lives in the exponent, so every method here is really one idea: get the exponents talking to each other, either by sharing a base or by taking a logarithm.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.