power law of logarithms

The third logarithm law is the most useful for solving equations: it lets you pull an exponent down and place it out in front as a multiplier. Where the product law handled multiplication and the quotient law handled division, the power law handles raising to a power, demoting it to ordinary multiplication.

The formula is log_b(M^p) = p · log_b(M), for any allowed base b, any positive M, and any real exponent p. You can see why: M^p is M multiplied by itself p times, the product law would add log_b(M) that many times, and adding the same thing p times is just multiplying by p. So the exponent slides to the front.

This is the single most important move for solving exponential equations. If an unknown is stuck in an exponent, like in 3^x = 20, taking a log of both sides lets the power law bring the x down to where you can isolate it. A common slip: the exponent p that comes out front applies to the whole argument M, not to anything outside the log.