Logarithm is undefined for non-positive values. Base must be positive and not equal to 1.
A logarithm answers the question: "To what power must we raise the base to get this number?" If b^y = x, then log_b(x) = y. Logarithms are the inverse operation of exponentiation.
Written as log(x) or log10(x), this is the most commonly used logarithm in everyday applications.
log(100) = 2 because 10^2 = 100 log(1000) = 3 because 10^3 = 1000
Written as ln(x), where e is approximately 2.71828. Natural logarithms are fundamental in calculus and many scientific applications.
ln(e) = 1 because e^1 = e ln(e^2) = 2 because e^2 = e^2
Written as log2(x), commonly used in computer science and information theory.
log2(8) = 3 because 2^3 = 8 log2(256) = 8 because 2^8 = 256
To calculate a logarithm with any base using a calculator that only has log or ln:
log_b(x) = log(x) / log(b) = ln(x) / ln(b)
pH scale (acidity), Richter scale (earthquakes), decibels (sound), and radioactive decay all use logarithmic scales.
Compound interest calculations, growth rates, and the rule of 72 use logarithms.
Algorithm complexity (Big O notation), binary search, and data compression rely on logarithms.
Log transformations help normalize skewed data and are used in logistic regression.