Introduction to Logarithms

What is a Logarithm?

A logarithm answers the question: “What exponent do I need?”

You know that $2^3 = 8$.

But what if I ask: “2 to what power gives 8?”

The answer is 3. A logarithm is how we write that question:

$\log_2(8) = 3$

The logarithm finds the exponent.

Converting Between Forms

These say the same thing:

How to Read It

$\log_2(8) = \text{ ?}$

Ask yourself: “2 to what power gives 8?”

$2^? = 8$ → $2^3 = 8$ → Answer: 3

More examples:

• $\log_3(9) = 2$ because $3^2 = 9$
• $\log_{10}(1000) = 3$ because $10^3 = 1000$
• $\log_5(1) = 0$ because $5^0 = 1$

Two Special Bases

Two bases are so common they get their own notation:

Common logarithm (base 10):

$\log(x) = \log_{10}(x)$

Natural logarithm (base e):

$\ln(x) = \log_e(x)$

What is e?

$e \approx 2.71828...$ is called Euler’s number.

It appears whenever you have continuous growth.

Where e Comes From

Imagine you put $1 in a bank with 100% interest per year.

The more frequently you compound, the more you earn.

But there’s a limit. No matter how often you compound, you never exceed $e$.

$e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$

Why e Matters

$e$ isn’t arbitrary. It emerges naturally whenever growth is proportional to size:

• Population growth
• Radioactive decay
• Compound interest
• Cooling and heating

That’s why $\ln$ is called the “natural” logarithm.

The Inverse Relationship

Logarithms and exponentials undo each other.

$\log_b(b^x) = x$

$b^{\log_b(x)} = x$

Just like $\sqrt{x^2} = x$, or how $+5$ and $-5$ cancel.

Examples:

• $\log_2(2^5) = 5$
• $10^{\log_{10}(100)} = 100$
• $\ln(e^3) = 3$
• $e^{\ln(7)} = 7$