Converting Between Forms

Converting between logarithmic and exponential form means moving one statement between two costumes: log_b(y) = x and bˣ = y say the same thing, and the base stays the base in both.

Convert log₅(25) = 2 to exponential form: the base is 5, the exponent is the log's answer, 2, and the result is the log's input — 5² = 25. Now the reverse trip: convert 3⁴ = 81 to logarithmic form. The base is still 3; the exponent 4 becomes the log's answer; the result 81 becomes its input: log₃(81) = 4. The rewriting rule in one line: the log's output is the exponent, always. Every log identity, every log equation, and every mental evaluation runs through this conversion, which is why it drills before anything else does.