Into One Log

The three rules also work backwards, and then they combine logs into one. For positive x, a sum of logs with the same base is the log of a product: log_3(x) + log_3(4) = log_3(4x). A difference is the log of a quotient: log_3(x) - log_3(4) = log_3(x/4). And a number in front goes up as a power: 2log_3(x) = log_3(x^2).

Why bother? An equation with a single log is one move from solved: you raise the base to both sides, as in the last chapter. Combining is how longer log equations reach that point.

The rules only join logs with the same base. log_2(x) + log_3(x) cannot become one log, because every rule comes from powers of a single base. When a number stands in front, bring it up as a power first, then combine.