Converting Between Logarithmic and Exponential Form

Every log statement is hiding an exponent question. Log base 4 of 64 is 3 is just a fancy way of asking: 4 to what power gives 64? The answer is 3, so 4³ = 64. Those two equations, log base 4 of 64 = 3 and 4³ = 64, carry identical information. Nothing is lost or gained rotating between them, you're just choosing which part to isolate.

The base never moves when you rotate forms. In log base b of y = x, b stays the base; it just slides down to become the base of the exponential b^x = y. The output of the log, x, becomes the exponent. The number you took the log of, y, becomes the result of the exponentiation. Keep those three roles straight: base, exponent, result, and the rotation is automatic.

A bare log, written with no base shown, is a specific convention: it means base 10. Writing ln means base e, roughly 2.718, a totally different number. So log(100) = 2 but ln(100) is not 2. Mixing these up is a common error, so always check which symbol is on the page. This conversion skill is the hinge for everything that follows in this unit: once an unknown exponent is pulled down into an ordinary equation, you can use log laws to solve for variables trapped in exponents, which is where growth and decay problems live.