College Algebra / Unit 8 Alg 455 worked example, 60 to 90 seconds
Expanding and Condensing Logarithms
The idea
Expanding a logarithm rewrites one log of a product, quotient, and power as a sum and difference of simple logs; condensing runs the same rules in reverse to a single log.
In practice
Expand log₂(4x³/y): the quotient rule splits off − log₂(y); the product rule splits log₂(4x³) into log₂(4) + log₂(x³); the power rule finishes: 2 + 3·log₂(x) − log₂(y). Spot-check at x = 2, y = 8: the original is log₂(32/8) = log₂(4) = 2, and the expansion gives 2 + 3 − 3 = 2. Condensing reverses the parade — coefficients climb back up as powers, sums fold into products, differences into quotients — and ends at one log, which is the form equation-solving wants.
Where people slip
Expansion runs on products, quotients, and powers only; a sum inside a log stays put.
Your place on the trail
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