The inverse of exponential — used for decibels, earthquake magnitude, pH, and compressing large scales.
A logarithmic function is the inverse of an exponential function. " In other words, if b^y = x, then log_b(x) = y.
The two most common logarithms are the common logarithm (log base 10, written log(x)) and the natural logarithm (log base e, written ln(x)). The graph of a logarithmic function is a curve that rises quickly near x = 0 and then grows more and more slowly — it is the horizontal reflection of the exponential curve.
f(x) = a · log_b(x) + c
Vertical stretch/compression and reflection
Base of the logarithm (b > 0, b ≠ 1)
Vertical shift of the graph
Domain
(0, +∞) — only defined for positive x
Range
All real numbers: (-∞, +∞)
X-intercept
(1, 0) — since log_b(1) = 0 for any base b
Vertical Asymptote
x = 0 (the y-axis)
Inverse Relationship
log_b(bˣ) = x and b^(log_b x) = x
Change of Base
log_b(x) = ln(x) / ln(b) = log(x) / log(b)
f(x) = log(x)Common log (base 10): log(100) = 2, log(1000) = 3.
f(x) = ln(x)Natural log (base e): ln(e) = 1, ln(e²) = 2.
f(x) = log₂(x)Log base 2: log₂(8) = 3, log₂(1024) = 10.
Explore different bases and compare with the exponential curve
A logarithm is just an exponent. log_b(x) asks: "What exponent do I put on b to get x?" For example, log₂(8) = 3 because 2³ = 8. It is the tool that "undoes" exponentiation.