Introduction to Logarithms

Have you ever looked at an equation like $2^x = 32$ and wondered how to solve for $x$? A logarithm is the mathematical tool designed to do exactly that. It answers the question: "To what power must the base be raised to produce a given number?"

Exponents and Logarithms: Two Sides of the Same Coin

Logarithms and exponential functions are inverses of each other. Every exponential equation can be rewritten as a logarithmic equation, and vice versa.

The fundamental relationship is: $\log_b(a) = c \iff b^c = a$

Here is what each part means:

• $b$ is the base (the number being multiplied by itself).
• $c$ is the exponent (the power the base is raised to).
• $a$ is the argument (the result of the exponential expression).

When you read $\log_b(a) = c$, say out loud: "The power I need to raise $b$ to, in order to get $a$, is $c$."

Evaluating Logarithms

Let's look at how to evaluate a logarithmic expression.

Example: Evaluate $\log_2(32)$.

1. Set the expression equal to a variable: $\log_2(32) = x$.
2. Rewrite it in exponential form using the fundamental relationship: $2^x = 32$.
3. Ask yourself: "2 raised to what power equals 32?"
4. Since $2 \times 2 \times 2 \times 2 \times 2 = 32$, we know that $2^5 = 32$.
5. Therefore, $\log_2(32) = 5$.

Converting Between Forms

Being able to switch back and forth between exponential and logarithmic forms is a crucial skill in algebra.

Example: Rewrite $5^3 = 125$ in logarithmic form.

1. Identify the base ($b = 5$), the exponent ($c = 3$), and the result ($a = 125$).
2. Plug these into the logarithmic structure: $\log_b(a) = c$.
3. The logarithmic form is: $\log_5(125) = 3$.

Important Restrictions

When working with logarithms $\log_b(a)$, keep these rules in mind:

• The base $b$ must be strictly greater than 0 and cannot equal 1 ($b > 0, b \neq 1$).
• The argument $a$ must be strictly positive ($a > 0$). You cannot take the logarithm of zero or a negative number!