Expanding Logarithmic Expressions

The three log rules

• Product rule: $\log(ab) = \log a + \log b$
• Quotient rule: $\log(a/b) = \log a - \log b$
• Power rule: $\log(a^n) = n \log a$

Step-by-step: Expand $\log_3\!\left(\dfrac{x^2 y}{z^3}\right)$

Step 1 — Apply the quotient rule (top minus bottom):

$\log_3(x^2 y) - \log_3(z^3)$

Step 2 — Apply the product rule to the first term:

$\log_3(x^2) + \log_3(y) - \log_3(z^3)$

Step 3 — Apply the power rule to bring exponents in front:

$2\log_3 x + \log_3 y - 3\log_3 z$

Final answer: $2\log_3 x + \log_3 y - 3\log_3 z$.

Going the other direction (condensing)

You can also condense: $3\log_2 x - \log_2(x+1) = \log_2\!\left(\frac{x^3}{x+1}\right)$. Power rule first ($3\log x = \log x^3$), then quotient rule.

Numerical check

Let $x = 3, y = 2, z = 2$, base = 3:

• Original: $\log_3(9 \cdot 2 / 8) = \log_3(2.25) \approx 0.738$
• Expanded: $2\log_3 3 + \log_3 2 - 3\log_3 2 = 2 + 0.631 - 1.893 = 0.738$ ✓

Try it in the visualization

Adjust test values for $x$, $y$, $z$. Each rule is applied and highlighted in a different color. The numerical check verifies both sides are equal.