Logarithm Graph (y = a·log_b(x-h) + k) – Tutorial

Logarithm Graph: y = a·log_b(x-h) + k

This tool visualizes the graph and key properties of a logarithmic function. Logarithms are especially important because they are the inverse of exponential functions.

1) Domain (Most critical part)

The argument of the logarithm must be positive:

(x - h) > 0 ⇒ x > h

Therefore, the line x = h is a vertical asymptote. The function approaches this line but never crosses it.

2) Effect of each parameter

• b (base):
  • b > 1 → logarithm is an increasing function.
  • 0 < b < 1 → logarithm is a decreasing function.
• a: scales the graph vertically. If a < 0, the graph is reflected over the x-axis (swapping increasing/decreasing behavior).
• h: shifts the graph left or right. Defines the domain boundary: x > h.
• k: shifts the graph up or down.

3) Key reference points

Logarithmic functions have two very useful reference points:

• When x - h = 1: log_b(1)=0 → at x = h+1:

  (h+1, k)

• When x - h = b: log_b(b)=1 → at x = h+b:

  (h+b, k+a)

These two points help you quickly "anchor" the graph.

4) Intercepts (with x and y axes)

• Y-intercept (x=0): Can be calculated only if 0>h (i.e., x=0 is in the domain).
• X-intercept (y=0): Solve 0 = a·log_b(x-h) + k.
  log_b(x-h) = -k/a ⇒ x-h = b^{-k/a} ⇒ x = h + b^{-k/a} (a≠0).

5) What does this tool do?

• Automatically applies the domain restriction (x>h).
• Shows the asymptote x=h (dashed line).
• Generates a sample points table.
• Reports key points and intercepts where applicable.

Note: This tool uses numerical sampling to plot the graph. Since logarithm values can grow very large near the asymptote, points very close to x=h are deliberately offset slightly during sampling.