Converting Between Forms
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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.
The base never moves; everything else trades places around it.
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