Logarithmic Function Properties

A logarithmic function is the inverse of an exponential function. If we write y = log_b(x), we are asking: 'To what power must we raise the base b to get x?' Understanding the graphical characteristics of logarithmic functions — their shape, intercepts, asymptotes, domain, and range — is essential for working with them in algebra, calculus, and real-world modeling. The graph of y = log_b(x) has a distinctive curved shape that grows without bound but does so very slowly. Unlike exponential graphs that rise steeply, logarithmic graphs climb gently, which is why logarithms are so useful for representing quantities that span many orders of magnitude, such as the Richter scale for earthquakes, the decibel scale for sound, and the pH scale for acidity. Every logarithmic function with base b > 1 shares a set of key properties: a vertical asymptote at x = 0, a domain of all positive real numbers, a range of all real numbers, and an x-intercept at (1, 0). Knowing these properties allows you to sketch any logarithmic graph quickly and to understand transformations applied to it.

Key takeaways
  • Domain and Range
  • Vertical Asymptote
  • Intercepts
  • Increasing vs. Decreasing Behavior
  • Effect of the Base
  • Transformations

Domain and Range

The domain of y = log_b(x) is (0, ∞) — only positive real numbers are allowed as inputs, because you cannot take the logarithm of zero or a negative number (no real power of a positive base produces zero or a negative result). The range is (-∞, ∞), meaning the output can be any real number. For example, log_2(0.125) = -3, log_2(1) = 0, and log_2(8) = 3.

Vertical Asymptote

As x approaches 0 from the right, log_b(x) decreases without bound (→ -∞) when b > 1. The y-axis (the line x = 0) is a vertical asymptote. The graph gets infinitely close to the y-axis but never touches or crosses it. There is NO horizontal asymptote — the function grows (slowly) forever in both the positive and negative y directions.

Intercepts

Every logarithmic function y = log_b(x) passes through the point (1, 0), because log_b(1) = 0 for any valid base b (since b^0 = 1). This is the x-intercept. There is no y-intercept because x = 0 is not in the domain. Additionally, the graph passes through (b, 1) since log_b(b) = 1.

Increasing vs. Decreasing Behavior

When b > 1 (e.g., base 2, base 10, base e), the function is strictly increasing: as x increases, y increases. When 0 < b < 1 (e.g., base 1/2), the function is strictly decreasing. In both cases the graph is concave down (curves downward). The natural logarithm ln(x) = log_e(x) with e ≈ 2.718 is the most important case where b > 1.

Effect of the Base

Larger bases make the graph rise more slowly. For x > 1: log_2(x) > log_10(x) > log_100(x). All graphs still pass through (1, 0) and (b, 1). For x between 0 and 1: larger bases give less negative (closer to zero) outputs — the curves are ordered in reverse. The base controls the steepness of the rise.

Transformations

Standard transformations apply: y = log_b(x) + k shifts the graph vertically by k units. y = log_b(x - h) shifts it horizontally by h units (and moves the vertical asymptote to x = h). y = -log_b(x) reflects it over the x-axis. y = log_b(-x) reflects it over the y-axis, producing a graph defined on (-∞, 0). The stretch y = a·log_b(x) scales vertically by factor a.

The Key Equation

y=log⁡b(x)  ⟺  by=x,b>0,  b≠1,  x>0y = \log_b(x) \iff b^y = x, \quad b > 0,\; b \neq 1,\; x > 0
  • b — Base (dimensionless): The base of the logarithm; must satisfy b > 0 and b ≠ 1. Common choices are b = 10 (common log), b = e ≈ 2.718 (natural log), or b = 2 (binary log).
  • x — Argument (input) (dimensionless): The input to the logarithm; must be strictly positive. It represents the value whose logarithm we seek.
  • y — Logarithm (output) (dimensionless): The output of the function; the exponent to which b must be raised to produce x. Can be any real number.

Worked Example

Problem. Describe all key graphical properties of f(x) = log_3(x - 2) + 1, including domain, range, asymptote, and intercepts.

Domain: (2,∞),V.A.: x=2,x-int: (73,0)\text{Domain: }(2,\infty),\quad \text{V.A.: }x=2,\quad \text{x-int: }\left(\tfrac{7}{3},0\right)

Answer. Domain (2, ∞); Range (-∞, ∞); Vertical asymptote x = 2; X-intercept (7/3, 0); no y-intercept; strictly increasing; passes through (5, 2).

Figure
Diagram
yxO-3-1023123456(1, 0)(3, 1)(1/3,−1)V.A.: x=0y = log₂(x)Domain: (0,∞) Range: ℝ
Origins · 1 minJohn Napier and how this was discovered
JN
John Napier
1550–1617 · Scottish

John Napier invented logarithms around 1614 and published his discovery in 'Mirifici Logarithmorum Canonis Descriptio.' His motivation was purely practical: to reduce the laborious multiplication and division of large numbers (especially in astronomy and navigation) to simple addition and subtraction using log tables. Though his original logarithms differed from modern ones, the concept was revolutionary. Napier was also known in his time as a theologian and inventor of military devices. His neighbors reportedly believed he was a sorcerer — partly because he used a black rooster to catch a thieving servant, claiming the rooster would identify the guilty party by touch.

  • Exponential Function Properties — Logarithmic functions are the inverses of exponential functions; their graphs are reflections of each other across y = x.
  • Inverse Functions — The logarithm is defined as the inverse of exponentiation; understanding inverse functions explains the domain/range swap and reflection symmetry.
  • Laws of Logarithms — Properties like log(ab) = log a + log b follow directly from the graphical and algebraic behavior of logarithmic functions.
  • Transformations of Functions — Shifts, reflections, and stretches of y = log_b(x) change the asymptote, intercepts, and domain in predictable ways governed by transformation rules.
  • Change of Base Formula — The formula log_b(x) = ln(x)/ln(b) connects logarithms of different bases and explains why all log graphs have the same shape, scaled by a constant factor.