Simplify log(a²b³) - 2log(ab)

Question

Simplify $\log(a^2b^3) - 2\log(ab)$ .

Solution — Step by Step

\log(a^2b^3) = \log(a^2) + \log(b^3)

Applying the power rule to each term:

= 2\log a + 3\log b

2\log(ab) = 2[\log a + \log b] = 2\log a + 2\log b

\log(a^2b^3) - 2\log(ab) = (2\log a + 3\log b) - (2\log a + 2\log b)

= 2\log a - 2\log a + 3\log b - 2\log b

= 0 + \log b

= \mathbf{\log b}

Why This Works

We applied three log laws:

Product rule: $\log(XY) = \log X + \log Y$
Power rule: $\log(X^n) = n\log X$
Subtraction: Collect like terms as in ordinary algebra

The answer $\log b$ has a clean interpretation: the original expression is equivalent to $\log\left(\frac{a^2b^3}{(ab)^2}\right) = \log\left(\frac{a^2b^3}{a^2b^2}\right) = \log b$ . We can verify this by combining the steps into one.

Alternatively, handle the subtraction as a single quotient first: $\log(a^2b^3) - \log(ab)^2 = \log\frac{a^2b^3}{a^2b^2} = \log b$ . This one-line approach is faster in exams once you’re confident with logs.

Alternative Method

Combine into a single logarithm first:

\log(a^2b^3) - 2\log(ab) = \log(a^2b^3) - \log(ab)^2

= \log\left(\frac{a^2b^3}{a^2b^2}\right) = \log\left(\frac{b^3}{b^2}\right) = \log b

Same answer, fewer steps. The quotient rule does the work directly.

Common Mistake

A common error: students write $2\log(ab) = 2\log a \cdot 2\log b$ by “distributing” the 2 as if it were multiplication. The correct expansion is $2[\log a + \log b] = 2\log a + 2\log b$ . The coefficient 2 multiplies the entire log, not just one part. Treat $2\log(ab)$ as $2 \times [\log a + \log b]$ using the product rule first.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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