Solving Exponential Equations

Addition undoes subtraction, multiplication undoes division, and logarithms undo exponents. When x is stuck in the exponent, no amount of adding or dividing will free it, because those operations don't touch exponents. A logarithm is the one tool built to reverse exponentiation, so it is the only move that actually reaches x where it lives.

The mechanism rests on one log property: log(b^x) = x times log(b). Once you take the log of both sides of an equation, any exponent attached to the variable's base drops down and becomes a multiplier instead. That turns an exponential equation into a linear one. So to solve 3 times 2^x = 96, first divide by 3 to isolate the power: 2^x = 32. Take log of both sides: log(2^x) = log(32), which becomes x times log(2) = log(32), so x = log(32)/log(2) = 5. Graphically, this is the x-coordinate where y = 3(2^x) crosses y = 96.

For 5^x+1 = 12, take log of both sides first: (x+1) times log(5) = log(12), so x + 1 = log(12)/log(5), giving x = log(12)/log(5) - 1 exactly, or about 0.544 as a decimal. Inequalities work the same way with one catch: if you ever multiply or divide by a negative number while isolating things, flip the inequality sign, same as always. Taking a log itself never flips anything here, since log is increasing for positive inputs. This topic is the algebraic payoff of everything logs set up earlier in the unit, and it is exactly what is needed to fit exponential models to real data later.