Math
Logarithm Graph (y = a·log_b(x) + c)
Enter coefficient a, base b, and shift c to graph y = a·log_b(x) + c. Automatically computes x-intercept and vertical asymptote. Only defined for x > 0.
Tips
- The logarithmic function is only defined for x > 0. The graph approaches the vertical asymptote x = 0 but never crosses it.
- When b > 1 the function increases (rises to the right). When 0 < b < 1 it decreases. A negative a flips this.
- Enter b ≈ 2.71828 to graph the natural logarithm y = ln(x). Enter b = 10 for the common logarithm y = log₁₀(x).
- The x-intercept is where y = 0, which occurs at x = b^(−c/a) (when a ≠ 0). It appears as a green dot on the graph.
FAQ
The real-valued logarithm is only defined for positive numbers. log(0) = −∞ (undefined) and log(x) for x < 0 produces a complex (non-real) number.
log₁(x) would mean solving 1^y = x, but 1^y = 1 for all y — so no solution exists for x ≠ 1. The base-1 logarithm is undefined.
The common logarithm uses base 10 — enter b = 10. The natural logarithm uses base e ≈ 2.71828 — enter b = 2.71828.
Side Note — Logarithms in decibels and earthquake magnitude
Logarithms are embedded in everyday measurement. The decibel (dB) scale for sound uses the base-10 logarithm: every 10 dB increase represents a 10× increase in sound energy. A sound jumping from 30 dB to 60 dB feels moderately louder but carries 1,000 times more energy.
The Richter scale for earthquakes is also logarithmic — a difference of 1 magnitude unit corresponds to roughly 31.6× more seismic energy. Logarithms are indispensable whenever a measurement spans many orders of magnitude.