How to Use Properties of Logarithms? (+FREE Worksheet!)
Properties of Logarithms, Example 4:
Exercises for Properties of Logarithms
Expand the logarithm.
2. \(\color{blue}{log_{a}{(\frac{1}{2})}=}\)
Free tool: try the problems from this lesson in our Logarithm Calculator.
Tutor-style math help
Use Properties of Logarithms: what to notice and how to work it
Logarithms skill
Logarithm properties let you expand one log into pieces or condense several logs into one. Each property comes from an exponent rule.
What to notice first
Match the structure inside the logarithm: products become sums, quotients become differences, and powers move to the front.
Common student mistake
Do not split a sum or difference inside one logarithm. \(\log_b(M+N)\) does not become \(\log_b M+\log_b N\).
Key formulas and cues
\(\log_b(MN)=\log_b M+\log_b N\)
\(\log_b\left(\frac MN\right)=\log_b M-\log_b N\)
\(\log_b(M^p)=p\log_b M\)
\(M>0,\ N>0,\ b>0,\ b\ne 1\)
A reliable path
- Translate firstAsk: the base to what power gives the input?
- Use rules legallyProducts, quotients, and powers have rules; sums do not split.
- Protect the domainKeep the log input positive and track asymptotes when graphing.
Worked examples
Expand a log
Example: \(\log_2(8x^3)\)
- Product rule separates 8 and x cubed.
- Power rule moves the 3 to the front.
- Keep the same base.
Answer: \(\log_2 8+3\log_2 x\)
Condense logs
Example: \(\log x+\log 5-\log 2\)
- A sum of logs becomes a product.
- A difference becomes a quotient.
- Write one logarithm.
Answer: \(\log\left(\frac{5x}{2}\right)\)
Try one before moving on
Try: Expand \(\log_3\left(\frac{x^2}{7}\right)\).
Answer: \(2\log_3 x-\log_3 7\).
Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.
x
Use Properties of Logarithms: pop-up practice
Answer these quick questions, then use the feedback to decide which part of the lesson to review.
Choose an answer to begin.
1. \(\log_b(MN)\) equals:
2. \(\log_b(M^4)\) equals:
3. \(\log_b(M+N)\) can be split with the product rule.
3. \(\color{blue}{log_{a}{(2^5×8)}}\)
4. \(\color{blue}{log_{b}{(2x×7y)}}\)
Condense into a single logarithm.
5. \(\color{blue}{log_{a}{x}+log_{a}{y}}\)
6. \(\color{blue}{log_{a}{2x}-2log_{a}{y}}\)
- \(\color{blue}{-log 5}\)
- \(\color{blue}{-log_{a} {2}}\)
- \(\color{blue}{5log_{a} {2}+log_{a} {8}}\)
- \(\color{blue}{log_{b} {2x}+log_{b} {7y}}\)
- \(\color{blue}{log_{a} {xy}}\)
- \(\color{blue}{log_{a} {\frac{2x}{y^2}}}\)
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