Logarithms: Definition and Conversion

By the end of this lesson, you will be able to convert between exponential and logarithmic form, evaluate logarithms without a calculator, identify the domain and range of logarithmic functions, and understand $\log$ and $\ln$ notation.

The Big Idea

A logarithm answers the question: "to what power must I raise the base to get this number?" Logarithms are the inverse functions of exponentials — the exact relationship that lets us solve equations like $10^x = 1000$ or $e^x = 20$. Without logarithms, equations with the variable in the exponent are unsolvable by algebraic means.

Definition

$$\log_b x = y \iff b^y = x$$

Read $\log_b x$ as "log base $b$ of $x$."

The base $b$ must satisfy: $b > 0$ and $b \neq 1$.

The argument $x$ must be positive: $x > 0$.

Special Bases

Common logarithm (base 10): $\log x = \log_{10} x$

Natural logarithm (base $e$): $\ln x = \log_e x$, where $e \approx 2.71828...$

These two are computed directly on any scientific calculator.

Converting Between Forms

Exponential to logarithmic:

$2^5 = 32 \iff \log_2 32 = 5$

$10^3 = 1000 \iff \log 1000 = 3$

$e^2 = x \iff \ln x = 2$

Logarithmic to exponential:

$\log_3 27 = 3 \iff 3^3 = 27$

$\ln x = 4 \iff e^4 = x$

Example 1 — Evaluating Logarithms

$\log_2 64$: $2^? = 64 = 2^6$ → answer: $6$

$\log_5 \frac{1}{25}$: $5^? = \frac{1}{25} = 5^{-2}$ → answer: $-2$

$\log_7 1$: $7^? = 1 = 7^0$ → answer: $0$

$\log_b b$: $b^? = b = b^1$ → answer: $1$

General rules: $\log_b 1 = 0$ and $\log_b b = 1$ for any valid base.

Domain and Range of Logarithmic Functions

$f(x) = \log_b x$:

• Domain: $(0, \infty)$ — argument must be positive
• Range: $(-\infty, \infty)$ — all reals
• $x$-intercept: $(1, 0)$ (since $\log_b 1 = 0$)
• Vertical asymptote: $x = 0$ (the $y$-axis)

Inverse Relationship

$f(x) = b^x$ and $g(x) = \log_b x$ are inverses.