Logarithmic Functions - Graph, Properties, Examples

What Is a Logarithmic Function?

A logarithmic function is a function of the form:

$$f(x) = \log_b(x)$$

where $b$ is the base ($b > 0, b \neq 1$). It's defined by the inverse relationship:

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

In words: $\log_b(x)$ is "the exponent you raise $b$ to in order to get $x$."

The two most-used logarithmic functions:

• Common logarithm: $\log(x) = \log_{10}(x)$ — base 10. Often written without a subscript.

• Natural logarithm: $\ln(x) = \log_e(x)$ — base $e \approx 2.71828$. The default in calculus.

The Graph of a Logarithmic Function

For $f(x) = \log_b(x)$ with $b > 1$:

• Passes through $(1, 0)$ — because $\log_b(1) = 0$ for any base.

• Passes through $(b, 1)$ — because $\log_b(b) = 1$.

• Domain: $(0, \infty)$ — only positive inputs.

• Range: $(-\infty, \infty)$ — all real outputs.

• Vertical asymptote at $x = 0$ — the curve approaches the y-axis but never touches it. As $x \to 0^+$, $f(x) \to -\infty$.

• Increasing for $b > 1$ — grows monotonically but very slowly. For example, $\log_{10}(10) = 1$ but $\log_{10}(10000) = 4$.

• Concave down — the curve bends downward as $x$ increases.

For $0 < b < 1$, the graph flips vertically — decreasing instead of increasing, but with the same domain/range/asymptote.

Properties of Logarithmic Functions

The behaviour of $\log_b$ is governed by the same identities that hold for logarithms in general (see Properties of Logarithms).

Algebraic Properties

$$\log_b(xy) = \log_b(x) + \log_b(y) \quad \text{(product rule)}$$

$$\log_b!\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \quad \text{(quotient rule)}$$

$$\log_b(x^n) = n \log_b(x) \quad \text{(power rule)}$$

$$\log_b(1) = 0, \quad \log_b(b) = 1$$

Function-Specific Properties

$$f(x \cdot y) = f(x) + f(y) \quad \text{(turns multiplication into addition)}$$

$$f(x^n) = n \cdot f(x) \quad \text{(scales by exponent)}$$

$$f^{-1}(x) = b^x \quad \text{(inverse is exponential)}$$

Calculus Properties (for $f(x) = \ln(x)$)

$$\frac{d}{dx} \ln(x) = \frac{1}{x}, \quad x > 0$$

$$\int \ln(x) , dx = x \ln(x) - x + C$$

The natural log is the unique logarithm whose derivative is $1/x$ — which is why $\ln$ is the default in calculus.

Three Worked Examples — Quick, Standard, Stretch

Quick — Evaluate a Log

Compute $\log_2(32)$.

Ask: $2^? = 32$. Since $2^5 = 32$, the answer is $\log_2(32) = 5$.

Standard — Solve a Log Equation

Solve $\log_3(x) = 4$ for $x$.

By definition: $\log_3(x) = 4 \iff 3^4 = x$. So $x = 81$.

Check. $\log_3(81) = 4$ because $3^4 = 81$ ✓.

Stretch — Graph Transformation

Sketch $f(x) = \log_2(x - 3) + 1$.

Start from $y = \log_2(x)$:

• Replace $x$ with $x - 3$ → shifts the graph right by 3 units. Vertical asymptote moves from $x = 0$ to $x = 3$.

• Add $1$ → shifts the graph up by 1 unit.

Key points:

• Original $(1, 0)$ moves to $(4, 1)$.

• Original $(2, 1)$ moves to $(5, 2)$.

Domain: $(3, \infty)$. Vertical asymptote: $x = 3$.

Why Do Logarithmic Functions Matter? (The Real-World GROUND)

"Logarithms shorten the labours of the astronomer." — Pierre-Simon Laplace.

Logarithmic functions are not abstract — they describe how the natural world compresses and expands across enormous scales.

• Richter scale (earthquakes). Magnitude is logarithmic — each step up represents 10× the seismic energy. $\log$ turns enormous ranges into readable numbers.

• Decibel scale (sound). Loudness in decibels is $L = 10 \log_{10}(I/I_0)$ — a logarithmic compression of a 12-order-of-magnitude range of intensities into the 0–120 dB human range.

• pH scale (acidity). $\text{pH} = -\log_{10}[H^+]$. A change of 1 pH unit = 10× change in acidity.

• Information theory. The information content of a message is $-\log_2(p)$ — measured in bits. Shannon's 1948 formulation defined the modern concept of information using logarithms.

• Population growth (early phase). Exponential growth $N(t) = N_0 e^{rt}$ has its natural inverse — $\ln(N/N_0)/r = t$ — which tells you when a population reaches a target size.

The logarithmic function was invented in 1614 by John Napier specifically to make multiplication of large numbers easier — astronomers were spending months on hand calculations that could be reduced to addition by using log tables. Slide rules — used for engineering calculation through the 1970s — were physical implementations of logarithmic functions.

Learn more: Properties of Logarithms

A Worked Example — Wrong Path First

Solve $\log_2(x + 1) = 3$.

The intuitive (wrong) approach. A student takes the logarithm definition the wrong way — "so $x + 1 = 2 \cdot 3 = 6$, giving $x = 5$."

Why it fails. The student confused $\log_b(y) = k$ with $y = bk$. The correct relationship is $y = b^k$ — exponential, not linear.

The correct method. By definition: $\log_2(x + 1) = 3 \iff x + 1 = 2^3 = 8$, so $x = 7$.

Check. $\log_2(7 + 1) = \log_2(8) = 3$ ✓.

What Are the Most Common Mistakes With Logarithmic Functions?

Mistake 1: Taking the log of a non-positive number

Where it slips in: Writing $\log_2(0)$ or $\log_2(-5)$ as if they were defined.

Don't do this: $\log_2(0) =$ anything.

The correct way: $\log_b(x)$ is undefined for $x \le 0$. The domain of every logarithmic function is $(0, \infty)$.

Mistake 2: Confusing $\log(x^2)$ with $(\log x)^2$

Where it slips in: Treating the two as equal because the notation looks similar.

Don't do this: $\log_2(x^2) = (\log_2 x)^2$.

The correct way: $\log_2(x^2) = 2 \log_2(x)$ by the power rule. $(\log_2 x)^2 = \log_2(x) \cdot \log_2(x)$ — entirely different.

Mistake 3: Forgetting the asymptote

Where it slips in: Sketching the graph of $\log_b(x)$ as crossing the y-axis.

Don't do this: Drawing the curve through $(0, 0)$.

The correct way: The y-axis is a vertical asymptote — the curve approaches it but never touches it. The graph passes through $(1, 0)$, not $(0, 0)$.

Key Takeaways

• A logarithmic function $f(x) = \log_b(x)$ is the inverse of $b^x$ — they're reflections of each other across $y = x$.

• Domain: $(0, \infty)$; range: $(-\infty, \infty)$; passes through $(1, 0)$.

• Vertical asymptote at $x = 0$ — the curve never crosses the y-axis.

• Algebraic rules: product → sum, quotient → difference, exponent → coefficient.

• The natural log $\ln$ is the unique log whose derivative is $1/x$ — the default in calculus and physics.

A Practical Next Step

Try these three before moving on to exponential growth and decay.

1. Compute $\log_5(125)$.

2. Solve $\log_2(x) = 5$ for $x$.

3. Sketch the graph of $f(x) = \ln(x + 2)$. What's the vertical asymptote?

If problem 3 felt tricky, $\ln(x + 2)$ shifts the natural log graph left by 2 — vertical asymptote at $x = -2$. Want a Bhanzu trainer to walk through more logarithmic-function problems? Book a free demo class — online globally.