Logarithm Calculator
Calculate logarithms of any base, solve log equations, and explore log properties—all in one tool.
Logarithm Calculator
Logarithmic Equation Solver
Logarithm Properties & Examples
Basic Properties
- log_b(1) = 0
- log_b(b) = 1
- log_b(xy) = log_b(x) + log_b(y)
- log_b(x/y) = log_b(x) - log_b(y)
- log_b(x^k) = k·log_b(x)
- log_a(x) = log_b(x) / log_b(a) (change of base)
Logarithm Examples
log10(100) = 2 because 10^2 = 100
ln(e) = 1 because e^1 = e
log2(16) = 4 because 2^4 = 16
Solving Logarithmic Equations
log_b(x) = y → x = b^y
log(x) = 3 → x = 10^3 = 1000
log(x) = 3 → x = 10^3 = 1000
Logarithm Function Visualization
Visualize any logarithm function logb(x):
Base:
Range (x):
to
Learn: What is a Logarithm?
A logarithm is the inverse operation to exponentiation. It finds the exponent y such that b^y = x for given b and x. Written as log_b(x) = y.
- Common logarithm is with base 10, written as log(x) or log10(x).
- Natural logarithm is with base e ≈ 2.718, written as ln(x).
- Binary logarithm has base 2, written as log2(x). Useful in computer science!
Real-world uses: Logarithms appear in sound intensity (decibel scale), earthquakes (Richter scale), pH (chemistry), investments (growth rates), radioactive decay, and more!
Useful Tip:
log_b(x) = \frac {ln(x)} {ln(b)} (change of base with natural log)
log_b(x) = \frac {ln(x)} {ln(b)} (change of base with natural log)
Further Exploration
- Explore logarithmic growth in population models, algorithms, and sound.
- Use log rules in simplifying complex equations or solving for unknowns.
- Apply natural log to exponential decay/growth, half-life, and compound interest.
For more, see resources like Wikipedia: Logarithm or math guides!