log(5x) is log 5 + log x, log(x ÷ 5) is log x − log 5, and log(x⁵) is 5 log x. These free printable worksheets show one of the three used where another belonged, and ask which answer is right.
The three laws are only memorable against each other. A sheet that drills one teaches a student to apply whichever they last saw, so all three appear here and the correct answer is a sum on one question, a difference on the next and a coefficient on the one after — which means the shape of the answer never gives it away.
Why this sheet works
The wrong answers are the confusions that actually happen: the logs multiplied because the numbers inside were, the two subtracted the wrong way round, the power applied to the log instead of brought down, and the product inside turned into a sum. None is a slip of the pen; each is a rule reached for in the wrong place.
How this one works
One question from this sheet, worked through a step at a time.
Spot the mistake: log(x⁴) + log(x⁵) = 20(log x)². Which answer is right?
log(x⁴) + log(x⁵)
log(x⁴) + log(x⁵)=4 log x + 5 log x
log(x⁴) + log(x⁵)=4 log x + 5 log x
log(x⁴) + log(x⁵)=9 log x
log(x⁴) + log(x⁵)=9 log x
they wrote:20(log x)²
Do not check by reading their line. Somebody else's working is the most persuasive thing on the page, and rereading it is how you end up agreeing with it — do the step yourself first, then compare.
On your own, and writing the step nobody writes down: log(x⁴) + log(x⁵) becomes 4 log x + 5 log x.
Tidy that up and it is 9 log x.
Now compare. They wrote 20(log x)², and the two logs were multiplied once their powers had come down — so the correct answer is 9 log x.
Read each line of working and choose the answer that is correct.
1Spot the mistake: log(x⁴) + log(x⁵) = 20(log x)². Which answer is right?log(x⁴ + x⁵)20 log x20(log x)²9 log xWhy?
2Check this line: log(3x) − log 3 = log x. Which is correct?log xlog 3 − log xlog x ÷ log 33 log xWhy?
3Check this line: log(x²) + log(x⁵) = 7 log x. Which is correct?log(x² + x⁵)10 log x7 log x10(log x)²Why?
4Check this line: log(14x) − log 14 = 14 log x. Which is correct?log x ÷ log 14log x14 log xlog 14 − log xWhy?
Algebra reasoning · Logarithm lawsAnswer key
1. log(x⁴) + log(x⁵) = 9 log x, not 20(log x)² (the two logs were multiplied once their powers had come down)2. log(3x) − log 3 = log x, which is what was written — this line is correct3. log(x²) + log(x⁵) = 7 log x, which is what was written — this line is correct4. log(14x) − log 14 = log x, not 14 log x (the multiplier inside the log was brought down as a power)
