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
Solving Logarithmic Equations
log(x) = 3 → x = 10^3 = 1000
Logarithm Function Visualization
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!
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!
Logarithm Calculator – Instantly Solve Log & ln Equations with Steps
Need to simplify a complex math equation involving logarithms? Whether you’re a student, engineer, or data analyst, our Logarithm Calculator makes solving logarithmic functions quick and easy. From base 10 to natural logs (ln), it handles all your exponential needs with step-by-step breakdowns and support for any base.
🧠 What Is a Logarithm?
A logarithm is the inverse operation of exponentiation. It tells you what power a base number must be raised to, in order to get another number.
For example:log₁₀(100) = 2 because 10² = 100log₂(8) = 3 because 2³ = 8
Common types of logarithms:
- 🔟 Common logarithm (log) – base 10
- 🧮 Natural logarithm (ln) – base e (≈ 2.718)
- 🔢 Binary log – base 2 (used in computer science)
✨ Key Features of the Logarithm Calculator
- 🔢 Solve for any custom base (not just 10 or e)
- 🧠 Show step-by-step solutions
- ✅ Supports natural log (ln), base 10 (log), and binary log
- 🔄 Convert between exponential & logarithmic forms
- 🧮 Handles advanced expressions with variables
📚 Logarithm Formula Reference
| Type | Formula | Example |
|---|---|---|
| Common Log (base 10) | log(x) = log₁₀(x) | log(100) = 2 |
| Natural Log (ln) | ln(x) = logₑ(x) | ln(e²) = 2 |
| Change of Base | log_b(x) = log(x)/log(b) | log₂(32) = log(32)/log(2) = 5 |
| Exponential Form | b^y = x ⇒ log_b(x) = y | 3² = 9 ⇒ log₃(9) = 2 |
🔢 Example Calculation
Problem: log₂(64)
Step 1: Use the formula
log₂(64) = ?
Step 2: Recognize that 2⁶ = 64
✅ Answer: log₂(64) = 6
🎯 Who Should Use the Log Calculator?
- 👨🎓 Students in high school & college (algebra, precalc, calculus)
- 🧑🔬 Scientists using exponential data
- 👨💻 Programmers working with binary systems
- 📈 Data analysts applying logarithmic scales
- 🧠 Math teachers preparing exercises
🔥 Features Included
- Logarithm calculator
- Solve log equations
- ln calculator with steps
- Change of base calculator
- Natural log solver
- Binary log calculator
- Simplify log expressions
- Exponential to log converter
🧭 Related Tools on Desmos Calculators
- Exponential Growth Calculator – great for compounding and population models
- Scientific Calculator – advanced math operations including trig, roots, and more
- Percent Error Calculator – useful for scientific experiments
🔗 External Trusted Sources
✅ Final Thoughts
The Logarithm Calculator is a must-have for anyone working with exponential functions. Whether you’re solving for x in a math problem or decoding exponential trends in science or finance, this tool brings accuracy, speed, and clarity.