Mathematics
Logarithms
A logarithm is the inverse of an exponential. log_b(x) = y means b^y = x. The three rules (product, quotient, power) plus change-of-base let you simplify or solve almost any logarithmic expression.
How to Approach Logarithms
Identify the form
Is it a single log? A sum or difference? A log equation? Each form has a different technique.
Apply log rules
Product: log(xy) = log(x) + log(y). Quotient: log(x/y) = log(x) - log(y). Power: log(x^n) = n·log(x). Change of base: log_b(x) = log(x)/log(b).
Solve or simplify
For log equations, rewrite as exponentials and solve. For expressions, combine or expand until you reach the simplest form.
Frequently Asked Questions
What's the natural log?+
ln(x) is shorthand for log_e(x) — logarithm with base e ≈ 2.71828. It appears everywhere in calculus, physics, and finance because e^x is its own derivative.
Why can't I take log of a negative number?+
Real logarithms are only defined for positive arguments. log(-1) requires complex numbers (the answer involves iπ). For real-valued logs, x must be > 0.
When do I use change of base?+
When your calculator only has log (base 10) and ln (base e) but you need a different base. log_b(x) = ln(x)/ln(b).
Related Topics
More step-by-step guides in Mathematics and adjacent subjects.
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