ACT Preparation / Unit 12 ACT 1115 worked example, 60 to 90 seconds
Expanding and Condensing Logarithms
The idea
To expand a logarithm, work in three steps. First, use the quotient rule to split a fraction into a difference. Second, use the product rule to split each product into a sum. Third, use the power rule to bring each exponent down as a coefficient, rewriting roots as fractional exponents first. To condense, run the steps in reverse: coefficients become exponents, sums become products, and differences become quotients.
In practice
Worked example: expand log(x³√y / z²). Step one: log(x³√y) − log(z²). Step two: log x³ + log √y − log z². Step three: 3 log x + (1/2) log y − 2 log z. Condensing runs the other way: 2 ln a − ln b + 3 ln c becomes ln(a²c³/b).
Where people slip
Apply the power rule only to a power of the whole input: in log(3x²), the 2 belongs to x alone, giving log 3 + 2 log x, not 2 log 3x.
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