- product becomes addition
Worked example
Expand log3(9x).
Step by step
- Use the product law.
- Evaluate the number part if possible.
Final answer
2 + log3 x
Very common mistake: log(x + y) does not split into log x + log y. The product law works for multiplication, not addition.
- division becomes subtraction
Worked example
Expand log2(16/y).
Step by step
- Use the quotient law.
- Evaluate the number part.
Final answer
4 - log2 y
Product vs quotient: Multiplication inside a log becomes addition. Division inside a log becomes subtraction.
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- power moves to the front
Worked example
Expand log5(x4).
Step by step
- Bring the power to the front.
Final answer
4log5 x
The power comes out the front: It does not become loga x4 outside the logarithm. The whole point is that the exponent becomes a coefficient.
The reverse direction: You can also go backwards: addition becomes a product, subtraction becomes division, and a coefficient becomes a power.
Worked example
Condense 2log x + log y - log z into one logarithm.
Step by step
- Move the coefficient 2 back in as a power.
- Use product and quotient laws in reverse.
Final answer
log((x2y)/z)
Keep it as one log: When condensing, the final answer should be a single logarithm, not several separate ones.