Log Rules for Simplifying Expressions

The logarithm of a product equals the sum of the logarithms: that single fact, log_b(xy) = log_b(x) + log_b(y), is why log rules exist. The four essential properties, product, quotient, power, and change of base, let you rewrite any logarithmic expression into a simpler form. Use them to expand a single log into separate terms, or condense several logs into one. The rules are in the table below; worked examples follow, along with the mistakes that trip up Algebra II students, chemistry students using pH, and anyone checking a calculator by hand.

Rule Name	Formula	Example
Product Rule	log_b(xy) = log_b(x) + log_b(y)	log_2(32 × 8) = log_2(32) + log_2(8) = 5 + 3 = 8
Quotient Rule	log_b(x/y) = log_b(x) − log_b(y)	log_10(1000/10) = log_10(1000) − log_10(10) = 3 − 1 = 2
Power Rule	log_b(x^p) = p · log_b(x)	log_3(9^4) = 4 · log_3(9) = 4 · 2 = 8
Logarithm of 1	log_b(1) = 0	log_5(1) = 0
Logarithm of the Base	log_b(b) = 1	log_7(7) = 1
Inverse Property	b^(log_b(x)) = x and log_b(b^x) = x	10^(log_10(100)) = 100
Change of Base	log_b(x) = log_c(x) / log_c(b)	log_2(10) = log_10(10) / log_10(2) = 1 / 0.3010 ≈ 3.3219

The product rule says that the logarithm of a multiplication is the sum of the logarithms of each factor. For log_b(xy), you write log_b(x) + log_b(y). This works because logarithms turn multiplication into addition, the same insight that made historical log tables useful for manual calculation.

How to Apply It

To expand log_5(25 × 125), break it into log_5(25) + log_5(125). Since 25 = 5^2 and 125 = 5^3, the result is 2 + 3 = 5. Check: 25 × 125 = 3125, and 5^5 = 3125.

Failure Case

The product rule does NOT apply to addition. log_b(a + b) is not log_b(a) + log_b(b). That is the single most common mistake in log rules. If you see a plus sign inside the log, the product rule does not help.

The quotient rule is the subtraction version of the product rule: log_b(x/y) = log_b(x) − log_b(y). It turns division into subtraction.

How to Apply It

Simplify log_10(10000 / 100). Write log_10(10000) − log_10(100). Since 10000 = 10^4 and 100 = 10^2, you get 4 − 2 = 2. Verify: 10000 / 100 = 100, and 10^2 = 100.

Domain Reminder

The argument inside a log must be positive. For log_b(x/y), both x and y must be positive individually. You cannot take log of a negative number or zero in real-number mathematics.

The power rule moves an exponent from inside the log to the front as a multiplier: log_b(x^p) = p · log_b(x). This works for any real exponent p, including fractions and negative numbers.

How to Apply It

Simplify log_2(64^3). The power rule gives 3 · log_2(64). Since 2^6 = 64, log_2(64) = 6, so the result is 3 × 6 = 18. Check: 64^3 = 262144, and 2^18 = 262144.

Fractional and Negative Exponents

For log_10(√1000) = log_10(1000^(1/2)) = (1/2) · log_10(1000) = (1/2) × 3 = 1.5. For log_3(1/27) = log_3(27^(-1)) = (−1) · log_3(27) = −3.

The change of base formula lets you convert a logarithm from one base to another: log_b(x) = log_c(x) / log_c(b). Use it when your calculator only has log (base 10) and ln (base e), or when you need a base that is not built in.

How to Apply It

To find log_2(50) without a log₂ button, write log_2(50) = log_10(50) / log_10(2). log_10(2) ≈ 0.3010. log_10(50) is about 1.6990 (since 10^1.6990 ≈ 50). The division gives 1.6990 / 0.3010 ≈ 5.64. Estimate check: 2^5 = 32, 2^6 = 64, so 5.64 is plausible.

Notation Watch

ISO 80000-2:2019 defines lb = log₂, lg = log₁₀, ln = logₑ. But in computer science textbooks like CLRS, lg means log₂. On a TI-84 Plus CE, use the logBASE( function: logBASE(50,2). In Microsoft Excel and Google Sheets, use LOG(50,2). Always check which convention your source uses.

Expanding a logarithmic expression means rewriting a single log into a sum or difference of logs. Condensing does the opposite, combine multiple logs into one. Both rely on the product, quotient, and power rules.

Expand: Worked Example

Expand log_10((1000x^2) / (√y)). Write it step by step: log_10(1000x^2) − log_10(√y). Then expand the first term: log_10(1000) + log_10(x^2). Apply the power rule: log_10(1000) + 2·log_10(x). For the second term: log_10(y^(1/2)) = (1/2)·log_10(y). Final expanded form: 3 + 2·log_10(x) − (1/2)·log_10(y), since log_10(1000) = 3.

Condense: Worked Example

Condense 2·log_2(a) + 3·log_2(b) − log_2(c) into a single logarithm. Apply the power rule first: log_2(a^2) + log_2(b^3) − log_2(c). Then the product rule: log_2(a^2 · b^3) − log_2(c). Finally the quotient rule: log_2((a^2 · b^3) / c).

Check Your Work

After expanding or condensing, test with simple numbers. Pick a = 2, b = 4, c = 8 and compute both sides with a calculator. For the condensed form above, log_2((2^2 · 4^3) / 8) = log_2((4 · 64) / 8) = log_2(256/8) = log_2(32) = 5. The expanded form: 2·1 + 3·2 − 3 = 2 + 6 − 3 = 5. They match.

What Goes Wrong

Readers forget that log_b(x) − log_b(y) condenses to log_b(x/y), not log_b(x − y). Another error: trying to condense log_b(x) + log_b(y) into log_b(x + y). That is wrong, condensed form is log_b(xy).

No rule exists for log_b(a + b). The logarithm of a sum does not simplify, you cannot break log_b(a + b) into log_b(a) + something. This is the most common trap in logarithm properties.

What Does Not Work

log_b(a + b) is not log_b(a) + log_b(b). It is not log_b(a) · log_b(b). It is not anything simpler. If a calculator says log_10(20 + 30) = log_10(50) ≈ 1.6990, and you try log_10(20) + log_10(30) ≈ 1.3010 + 1.4771 = 2.7781, the two numbers are not equal. The product rule applies only to multiplication inside the argument.

How to Handle a Sum Inside a Log

If you encounter log_b(a + b), either evaluate it directly with a calculator or leave it as is. No algebraic transformation can separate the terms. This contrasts with logarithms of products, quotients, and powers, which have clean rules.

Related Traps

log_b(a^b) is not (log_b a)^b. The first means b · log_b(a); the second means the log result raised to a power. They are entirely different. Also, log_b(a) − log_b(b) is not log_b(a − b), it is log_b(a/b). Always check the operation inside the log.

Five mistakes appear so often in logarithm work that they deserve a separate warning.

Mistake 1: Confusing ln and log

ln(x) is log base e; log(x) is often base 10. Using ln in a pH calculation gives a wrong answer by a factor of about 2.3026, since ln(x) = log(x) · ln(10). pH = −log_10([H+]), not −ln([H+]).

Mistake 2: Applying the Product Rule to Addition

log_b(a + b) is not log_b(a) + log_b(b). This error appears in homework and on exams every semester. If the argument is a sum, stop. No rule applies.

Mistake 3: Forgetting the Domain

The argument of a log must be positive. log_2(0) is undefined; log_2(−4) is undefined in real numbers. Before applying any rule, check that every argument inside a log is greater than zero.

Mistake 4: Misapplying the Power Rule to a Product

log_b(xy)^p is not p · log_b(xy). The power rule applies to the argument, not to the log function itself. Write log_b((xy)^p) = p · log_b(xy) = p · (log_b(x) + log_b(y)).

Mistake 5: Richter Scale Energy Confusion

Each whole-number step on the Richter scale multiplies wave amplitude by 10, but energy release multiplies by about 31.6 (10^1.5). Saying a magnitude 6 earthquake is 10 times stronger than a magnitude 5 confuses amplitude with energy. The USGS publishes the correct factor.