Mathfolis

Logarithmic Function Manipulation

Unit 2 · Exponential and Logarithmic Functions

What AP Precalc asks here

Three core log rules let you rewrite expressions: the product rule log⁡(xy)=log⁡x+log⁡y\log(xy) = \log x + \log y, the quotient rule log⁡(x/y)=log⁡x−log⁡y\log(x/y) = \log x - \log y, and the power rule log⁡(xp)=plog⁡x\log(x^p) = p \log x. Run them forward to expand a single log into a sum/difference; run them backward to condense. The change-of-base formula lets you evaluate any log via your calculator's log⁡\log or ln⁡\ln key.

The three rules

Product
log⁡b(xy)=log⁡bx+log⁡by\log_b(xy) = \log_b x + \log_b y
Quotient
log⁡b ⁣(xy)=log⁡bx−log⁡by\log_b\!\left(\tfrac{x}{y}\right) = \log_b x - \log_b y
Power
log⁡b(xp)=plog⁡bx\log_b(x^p) = p \log_b x

Change of base

General
log⁡bx=log⁡cxlog⁡cb\log_b x = \frac{\log_c x}{\log_c b}
Caution: log⁡(x+y)≠log⁡x+log⁡y\log(x + y) \neq \log x + \log y. The rules apply to products, quotients, and powers — not sums.

AP problem types

Type 1

Expand a single log into a sum/difference

Apply quotient first to split top and bottom, then power on each factor with an exponent, then product on any remaining factors.

Example 1
Expand log⁡(x3yz2)\log\left(\dfrac{x^3 \sqrt{y}}{z^2}\right). (A) 3log⁡x+12log⁡y−2log⁡z3\log x + \tfrac{1}{2}\log y - 2\log z (B) 3log⁡x+2log⁡y−2log⁡z3\log x + 2\log y - 2\log z (C) log⁡(x3)−log⁡(z2)+log⁡(y)\log(x^3) - \log(z^2) + \log(\sqrt{y}) (D) log⁡3x+log⁡y−log⁡2z\log 3x + \log\sqrt{y} - \log 2z

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Type 2

Condense multiple logs into one

Run the rules in reverse: coefficients become exponents, sums become products, differences become quotients.

Example 2
Condense 2log⁡a+log⁡b−3log⁡c2 \log a + \log b - 3 \log c to a single log. (A) log⁡(a2bc3)\log\left(\dfrac{a^2 b}{c^3}\right) (B) log⁡(a2+bc3)\log\left(\dfrac{a^2 + b}{c^3}\right) (C) log⁡(a2bc−3)\log(a^2 b c^{-3}) — incorrect interpretation of coefficient (D) log⁡(2a+b−3c)\log(2a + b - 3c)

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Type 3

Change of base

Rewrite using log⁡bx=log⁡cx/log⁡cb\log_b x = \log_c x / \log_c b with c=10c = 10 or c=ec = e, then compute.

Example 3
Approximate log⁡350\log_3 50 to three decimal places. (A) ≈1.732\approx 1.732 (B) ≈3.561\approx 3.561 (C) ≈4.500\approx 4.500 (D) ≈11.510\approx 11.510

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