The properties of logarithms are rules for rewriting logs: the log of a product is a sum, logb(MN) = logbM + logbN; the log of a quotient is a difference; and the log of a power is a multiple, logb(Mp) = plogbM. They come from the rules of exponents.
In Spanish
propiedades de los logaritmos
Also called
log rules, laws of logarithms, logarithm rules, product rule, quotient rule, power rule
Every rule can be checked with numbers. Because a log is an exponent, multiplying inside the log becomes adding outside it, just as 23 · 22 = 25.
Formal definition and notation
For a base b > 0, b ≠ 1, and positive numbersM and N: logb(MN) = logbM + logbN, logb( MN ) = logbM − logbN, and logb(Mp) = plogbM for any real p. Also logb 1 = 0, logbb = 1, logb(bx) = x, and the change-of-base formula logbM = logaMlogab.
Each log rule mirrors an exponent rule: bm · bn = bm + n gives the product rule, and (bm)p = bmp gives the power rule.
Expanding uses the rules left to right: log2(8x3) = 3 + 3log2x. Condensing goes right to left.
There is no rule for the log of a sum: log (M + N) is not logM + logN.
The power rule lets you solve exponential equations: from 3x = 50, take logs to get x = log 50log 3 ≈ 3.56.
The rules hold for every allowed base, including base 10 and base e (Euler's number).
The logarithm properties
The logarithm properties
logb(MN) = logbM + logbN
logb( MN ) = logbM − logbN
quotient rule
logb(Mp) = plogbM
power rule
logbM = logMlogb
change of base, to use a calculator’s log or ln key
logb 1 = 0, logbb = 1
because b0 = 1 and b1 = b
Each rule needs M > 0 and N > 0: logs of zero or negative numbers are undefined in real numbers.
Try it: a log is an exponent
Interactive
Here 25 = 32, so log2 32 = 5. Change the exponent and watch the value: adding 1 to the exponent doubles the value, which is the product rule log2(2M) = 1 + log2M.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Properties of Logarithms examples, step by step
Example 1: Expand a logarithm
Warm-up
Expand log3(9x).
Product rule: log3(9x) = log3 9 + log3x.
32 = 9, so log3 9 = 2.
Result: 2 + log3x.
Answer:log3(9x) = 2 + log3x.
Example 2: Condense and evaluate
Core
Write 2log 5 + log 4 as a single logarithm and evaluate it (base 10).
The natural log ln is a logarithm with base e. All the same properties apply: ln (ab) = lna + lnb.
Quick check: 3 questions
Question 1 of 3
Which is equal to log5(25y)?
Show answer and explanation
Answer: B. Product rule: log5 25 + log5y = 2 + log5y, because 52 = 25. 25log5y misuses the power rule, 5 confuses the base with the log, and 2log5y multiplies instead of adding.
Question 2 of 3
What is log3 54 − log3 2?
Show answer and explanation
Answer: A. Quotient rule: log3( 542 ) = log3 27 = 3. 27 forgets to take the log, log3 52 subtracts the inputs, and the last choice divides the logs.
Question 3 of 3
Solve 4x = 20. Which expression gives x?
Show answer and explanation
Answer: D. Take log of both sides: xlog 4 = log 20 (power rule), so x = log 20log 4 ≈ 2.16. The other choices divide 20 by 4 in some form, which treats the exponent like a factor.
Properties of Logarithms: frequently asked questions
What are the three main properties of logarithms?
The product rule turns the log of a product into a sum, logb(MN) = logbM + logbN. The quotient rule turns the log of a quotient into a difference. The power rule brings an exponent down as a multiplier, logb(Mp) = plogbM. Together with change of base, they handle almost every log problem in algebra.
Why do the log properties work?
A logarithm is an exponent, so log rules are exponent rules in disguise. If logbM = m and logbN = n, then M = bm and N = bn, so MN = bm + n. Taking the log gives logb(MN) = m + n, which is the product rule. The other rules come the same way.
How do you change the base of a logarithm?
Divide the log of the number by the log of the base, using any base your calculator has: logbM = logMlogb = lnMlnb. For example, log2 10 = log 10log 2 ≈ 3.32. This is how you evaluate logs with bases other than 10 and e.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSF-BF.B.5, CCSS HSF-LE.A.4
Cite this entry
Effortless Math Team. “Properties of Logarithms.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/properties-of-logarithms/
Flashcard
TermProperties of Logarithmslogb(MN), logb( MN ), logb(Mp)
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