Logarithm Essentials: Definition, Properties, and Calculations

2026.04.24

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Logarithms are the inverse of exponentiation and one of the most frequently tested concepts in high school math. They go hand-in-hand with exponential functions and appear throughout algebra and calculus.

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Logarithms appear in 2–3 questions on most standardized math exams. Master the definition and the three key laws, and you can solve the vast majority of log problems.

What Is a Logarithm?

Definition

A logarithm answers the question: "How many times must I multiply the base to get this number?"

For $a > 0$ , $a \neq 1$ , and $N > 0$ , if $a^x = N$ then $x = \log_a N$ .

$\log_a N = x \iff a^x = N$

Here $a$ is the base and $N$ is the argument.

Conditions for a Valid Logarithm

A logarithm $\log_a N$ is defined only when:

Base condition: $a > 0$ and $a \neq 1$
Argument condition: $N > 0$
If either condition fails, the logarithm is undefined

Common Log vs Natural Log

Type	Base	Notation	Use
Common log	10	$\log N$ (base omitted)	Digit counting, standardized tests
Natural log	$e \approx 2.718$	$\ln N$	Calculus, science

Logarithm Properties and Laws

Step 1: Basic Properties

For any valid base $a$ :

$\log_a 1 = 0 \quad (\because a^0 = 1)$

$\log_a a = 1 \quad (\because a^1 = a)$

$a^{\log_a N} = N$

Step 2: The Three Laws of Logarithms

For $M > 0$ and $N > 0$ :

Law 1 — Product → Addition

$\log_a MN = \log_a M + \log_a N$

Law 2 — Quotient → Subtraction

$\log_a \frac{M}{N} = \log_a M - \log_a N$

Law 3 — Power → Multiplication

$\log_a M^k = k \log_a M$

Step 3: Change of Base Formula

The key formula for converting between different bases:

$\log_a b = \frac{\log_c b}{\log_c a}$

Useful variations:

$\log_a b = \frac{1}{\log_b a}$

$\log_a b \cdot \log_b c = \log_a c$

⚠️

The most common mistake is thinking

\log_a(M + N) = \log_a M + \log_a N

. The product rule only works for multiplication inside the log, not addition!

Advanced: Applications of Logarithms

Finding the Number of Digits

The number of digits in a positive integer $N$ is $\lfloor \log_{10} N \rfloor + 1$ .

For example, $2^{10} = 1024$ :

$\log_{10} 1024 \approx 3.01 \Rightarrow \lfloor 3.01 \rfloor + 1 = 4 \text{ digits}$

Practice Problems

Example 1: Applying Log Laws

Problem: Find the value of $\log_2 12 - \log_2 3$ .

Show Solution

Apply the quotient rule:

$\log_2 12 - \log_2 3 = \log_2 \frac{12}{3} = \log_2 4 = 2$

Answer: $2$

Example 2: Change of Base

Problem: Find the value of $\log_2 3 \times \log_3 8$ .

Show Solution

Use the chain rule for logarithms:

$\log_2 3 \times \log_3 8 = \log_2 8 = 3$

Alternatively, $\log_3 8 = 3\log_3 2$ , so:

$\log_2 3 \times 3\log_3 2 = 3 \times \log_2 3 \times \frac{1}{\log_2 3} = 3$

Answer: $3$

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When solving log problems, always start by unifying the bases and prime-factoring the arguments. Most problems are solved in these two steps.

Top 3 Common Mistakes

Ignoring the argument condition — Forgetting to check $N > 0$ in $\log_a N$ and substituting negative numbers or zero. Always verify the argument after solving.
Confusing addition with multiplication — $\log(M + N) \neq \log M + \log N$ . You can only separate logs when the argument is a product.
Swapping numerator and denominator in base change — In $\log_a b = \frac{\log b}{\log a}$ , $b$ is in the numerator and $a$ is in the denominator.

Related Concepts

Exponential Functions — inverse pair
Integration — log in calculus
Quadratic Equations — algebraic foundation

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