Logarithm Rules — Formulas, Table & Examples

Q: What is the difference between log, ln, and log base a?

$\log x$ (with no base written) means $\log_{10} x$, the common logarithm. $\ln x$ means $\log_e x$, the natural logarithm, where $e \approx 2.718$. $\log_a x$ is the general form with any valid base $a$. All three obey the same set of logarithm rules — only the base differs.

Q: Can you distribute a logarithm over addition or subtraction?

No. A very common error is writing $\log(x + y) = \log x + \log y$, but this is wrong. The product rule says $\log(xy) = \log x + \log y$ — multiplication inside becomes addition outside. There is no rule that simplifies $\log(x + y)$ further.

Logarithm Rules

Algebra rules used when working with logarithms.

For the following, assume that x, y, a, and b are all positive. Also assume that a ≠ 1, b ≠ 1.

Definitions

1. log_a x = N means that a^N = x.

2. log x means log₁₀ x. All log_a rules apply for log. When a logarithm is written without a base it means common logarithm.

3. ln x means log_e x, where e is about 2.718. All log_a rules apply for ln. When a logarithm is written "ln" it means natural logarithm.
Note: ln x is sometimes written Ln x or LN x.

Rules

1. Inverse properties: log_a a^x = x and a^(loga ^x) = x

2. Product: log_a (xy) = log_a x + log_a y

3. Quotient:

4. Power: log_a (x^p) = p log_a x

5. Change of base formula:

Careful!!

log_a (x + y) ≠ log_a x + log_a y

log_a (x – y) ≠ log_a x – log_a y

Key Formula

\text{Product: } \log_a(xy) = \log_a x + \log_a y

\text{Quotient: } \log_a\!\left(\frac{x}{y}\right) = \log_a x - \log_a y

\text{Power: } \log_a(x^p) = p\,\log_a x

\text{Change of Base: } \log_a x = \frac{\log_b x}{\log_b a}

\text{Inverse Properties: } \log_a(a^x) = x \quad\text{and}\quad a^{\log_a x} = x

Where:

$a$ = The base of the logarithm (must be positive and not equal to 1)
$b$ = A new base used in the change of base formula (positive, not equal to 1)
$x$ = A positive real number (the argument of the logarithm)
$y$ = A positive real number (a second argument)
$p$ = Any real number (the exponent in the power rule)

Worked Example

Problem: Use logarithm rules to expand and simplify:

\log_2\!\left(\dfrac{8x^3}{4}\right)

Step 1: Simplify the fraction inside the logarithm.

\log_2\!\left(\frac{8x^3}{4}\right) = \log_2(2x^3)

Step 2: Apply the product rule to split the logarithm of a product into a sum.

\log_2(2x^3) = \log_2 2 + \log_2(x^3)

Step 3: Evaluate

\log_2 2

using the inverse property (

\log_a a = 1

\log_2 2 = 1

Step 4: Apply the power rule to bring the exponent 3 in front of the logarithm.

\log_2(x^3) = 3\log_2 x

Step 5: Combine all parts to write the final expanded form.

1 + 3\log_2 x

Answer:

\log_2\!\left(\dfrac{8x^3}{4}\right) = 1 + 3\log_2 x

Another Example

This example works in the opposite direction — condensing separate logarithms into one — and uses common logarithms (base 10) instead of base 2.

Problem: Condense the expression

2\log 5 + \log 4

into a single logarithm and evaluate.

Step 1: Use the power rule in reverse to move the coefficient 2 back inside as an exponent.

2\log 5 = \log(5^2) = \log 25

Step 2: Apply the product rule in reverse to combine the two logarithms into one.

\log 25 + \log 4 = \log(25 \times 4) = \log 100

Step 3: Evaluate

\log 100

. Since

\log

means

\log_{10}

and

10^2 = 100

\log 100 = 2

Answer:

2\log 5 + \log 4 = 2

Frequently Asked Questions

What is the difference between log, ln, and log base a?

\log x

(with no base written) means

\log_{10} x

, the common logarithm.

\ln x

means

\log_e x

, the natural logarithm, where

e \approx 2.718

\log_a x

is the general form with any valid base

a

. All three obey the same set of logarithm rules — only the base differs.

Can you distribute a logarithm over addition or subtraction?

No. A very common error is writing

\log(x + y) = \log x + \log y

, but this is wrong. The product rule says

\log(xy) = \log x + \log y

— multiplication inside becomes addition outside. There is no rule that simplifies

\log(x + y)

further.

When do you use the change of base formula?

You use it when you need to evaluate a logarithm whose base is not available on your calculator. Most calculators only have

\log

(base 10) and

\ln

(base

e

) buttons. The change of base formula lets you convert any base:

\log_a x = \frac{\ln x}{\ln a}

\frac{\log x}{\log a}

Product Rule vs. Power Rule

	Product Rule	Power Rule
Formula	$\log_a(xy) = \log_a x + \log_a y$	$\log_a(x^p) = p\,\log_a x$
What it does	Splits a logarithm of a product into a sum of logarithms	Moves an exponent from inside the log to a coefficient outside
Reverse use	Combines $\log_a x + \log_a y$ into $\log_a(xy)$	Converts $p\,\log_a x$ into $\log_a(x^p)$
Common confusion	Students misapply it to sums: $\log(x+y) \neq \log x + \log y$	Students confuse $(\log x)^p$ with $\log(x^p) = p\log x$

Why It Matters

Logarithm rules appear throughout Algebra 2, Precalculus, and Calculus whenever you solve exponential equations, differentiate logarithmic functions, or work with scientific scales like pH or decibels. They are essential on the SAT, ACT, and AP exams. Mastering these rules also builds the foundation for understanding exponential growth, compound interest, and data transformations in statistics.

Common Mistakes

Mistake: Writing

\log_a(x + y) = \log_a x + \log_a y

, incorrectly distributing the logarithm over addition.

Correction: The product rule applies to multiplication, not addition.

\log_a x + \log_a y = \log_a(xy)

. There is no simplification rule for

\log_a(x + y)

Mistake: Treating

\frac{\log_a x}{\log_a y}

\log_a\!\left(\frac{x}{y}\right)

Correction: The quotient rule says

\log_a\!\left(\frac{x}{y}\right) = \log_a x - \log_a y

(subtraction, not division). Dividing two logarithms does not simplify using the quotient rule; it relates to the change of base formula instead.

Related Terms

Logarithm — The fundamental concept these rules apply to
Common Logarithm — Base-10 logarithm, the default when no base is shown
Natural Logarithm — Base-e logarithm, written as ln
Change of Base Formula — Converts between logarithm bases
e — The base of the natural logarithm, approximately 2.718
Exponent — Logarithms are the inverse of exponentiation
Algebra — The branch of math where these rules are studied
Exponential Function — The inverse function of a logarithmic function