How can I use the product rule for logarithms without changing the base?
Quick answer
For a valid base a and positive numbers M and N, use log_a(MN) = log_a(M) + log_a(N). Keep the same base on both logarithms. For example, log_2(8·4) = log_2(8) + log_2(4) = 3 + 2 = 5. This rule applies to multiplication inside the logarithm, not addition: log_2(1+1) = 1, while log_2(1) + log_2(1) = 0. [S1]
The question, the details, the next step.
What does the product rule say, and how do I make sure I don’t accidentally change the base?
The product rule says that a logarithm of a product can be written as a sum of logarithms:
log_a(MN) = log_a(M) + log_a(N)
The base stays the same: both logarithms on the right use the original base a. The rule requires a > 0, a ≠ 1, and positive arguments M and N. [S1]
A quick way to check your work is to look inside the original logarithm. If its arguments are multiplied, the product rule may apply. If they are added, this rule does not turn that addition into separate logarithms.
Could you show the rule with numbers and walk through each step?
Suppose you need to simplify log_2(8·4).
- Identify the product inside the logarithm: 8·4.
- Split it using the same base:
log_2(8) + log_2(4). - Evaluate each logarithm:
log_2(8) = 3because 2³ = 8, andlog_2(4) = 2because 2² = 4. - Add: 3 + 2 = 5.
So log_2(8·4) = 5. You can check against the original: 8·4 = 32, and 2⁵ = 32. This is the same-base product property described by OpenStax. [S1]
What if there’s a plus sign inside the logarithm? Can I split it the same way?
No. The product rule is for multiplication inside the logarithm; it does not give an expansion rule for addition. [S1]
For a clear counterexample, compare log_2(1+1) with the tempting but invalid split log_2(1) + log_2(1).
- Evaluate the original:
log_2(1+1) = log_2(2) = 1. - Evaluate the split expression:
log_2(1) + log_2(1) = 0 + 0 = 0.
The answers differ, so the split is not valid. The distinction is between the operation joining the arguments: a product permits this rule; a sum does not. This counterexample illustrates that distinction; it does not supply a replacement rule for logarithms of sums.
Can you give me a short checklist to use while solving a problem?
Use this check before splitting a logarithm:
- Check the base: Is it positive and not equal to 1?
- Check the arguments: Are the factors positive?
- Check the operation: Are the terms multiplied inside the logarithm, rather than added?
- Keep the base: Write the original base on every logarithm after splitting.
- Evaluate and verify: Work out each logarithm and, when practical, compare with the original expression.
Copyable setup:
log_[same base](positive factor × positive factor)
= log_[same base](first factor) + log_[same base](second factor)
For example: log_3(9·27) = log_3(9) + log_3(27) = 2 + 3 = 5. The product property supports the split when its base and argument conditions hold. It does not establish how to split a sum inside a logarithm. [S1]
A practical checklist
Confirm the base is positive and not 1.
Confirm each factor inside the logarithm is positive.
Check that the factors are multiplied, not added.
Keep the original base on every logarithm after splitting.
Evaluate the parts and check the result against the original when practical.
A few more useful answers
Does the base change when I use the product rule?
No. Both logarithms in the expanded expression use the original base. [S1]
Can I use the product rule if one of the factors is zero or negative?
Not under the stated real-logarithm conditions: the factors M and N must be positive. [S1]
Does the product rule tell me how to expand log_a(M+N)?
No. It applies to a product, not a sum inside the logarithm, and provides no expansion rule for that sum. [S1]
Sources
References used for this guide.
- [S1] OpenStax Intermediate Algebra 2e: Use the Properties of Logarithms openstax.org · Checked