Logarithmic Functions & Their Equations

Explore the general transformation equation and learn what each parameter (a, b, h, k) does to the parent logarithmic function. Master the mapping notation formula, understand how to find asymptotes and domains, and work through detailed investigations comparing different logarithmic functions.

Lesson 3.6 of Exponents & Logarithms in Math 30-1: Alberta curriculum lessons.

What you'll learn in this lesson

• The Transformation Equation & What Each Parameter Does. Explore the general transformation equation and learn what each parameter (a, b, h, k) does to the parent logarithmic function. Master the mapping notation formula, understand how to find asymptotes and domains, and work through detailed investigations comparing different logarithmic functions.
• Finding the Equation From a Graph. Learn the systematic strategy for reading a logarithmic graph and finding its equation. Work through a detailed example showing how to identify the vertical asymptote, determine each parameter step-by-step, and verify your answer using key points on the graph.
• Sketching Transformed Log Graphs: Full Examples. Master the complete process for sketching transformed logarithmic functions from scratch. Learn the three key parent points for any log base, apply mapping rules to transform points, identify intercepts, and work through detailed examples including y = −log₃(x + 7) + 3 and y = 2log(2x + 6) + 4.

Inside the lesson: a free preview

The General Form

Just like exponential functions, logarithmic functions can be transformed by stretching, reflecting, and shifting. All of those transformations are captured in one general equation:

Order of Transformations: SRT. The order you apply transformations matters. Always follow SRT unless the question tells you otherwise:

Stretches and reflections first → Translations second.

What Each Parameter Does

Parameter a: Vertical Stretch and Reflection. Vertically stretches the graph by a factor of |a| about the x-axis. If a < 0, the graph is also reflected over the x-axis. Every y-value on the parent function gets multiplied by a.

Parameter b: Horizontal Stretch and Reflection. Horizontally stretches the graph by a factor of (1)/(|b|) about the y-axis. If b < 0, the graph is also reflected over the y-axis. Every x-value on the parent function gets divided by b.

Parameter h: Horizontal Translation. Shifts the graph h units horizontally. The direction is opposite to the sign, if h = 3, the graph moves right 3. If h = -2, the graph moves left 2. This parameter is the only one that moves the vertical asymptote.

Parameter k: Vertical Translation. Shifts the graph k units vertically. If k = 4, every point moves up 4. The range stays y R regardless of k.

Worked example: Find the equation in the form y = a · _2(x - h) + k

Here is the graph:

Take a good look before reading the solution. Ask yourself: where is the vertical asymptote? Which direction does the curve go? Where does it cross the x-axis?

Reading the graph. The curve has a vertical asymptote at x = 1. The curve is to the right of x = 1, increases as x increases, and two points are clearly marked: (2, 0) and (5, 2).

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