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
- 01
Identify the form
Is it a single log? A sum or difference? A log equation? Each form has a different technique.
- 02
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).
- 03
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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