Calculate logarithms with any base, anti-logarithms, and solve logarithmic equations.
Four tabs: compute log base-n, anti-logarithm (exponential), solve logarithmic equations, and interactively verify 6 properties of logarithms. Reference table for log, ln, and log base 2 values.
Calculator information
📋 How to use this calculator
- Choose the "Compute Log" tab to compute log_b(x); enter the base b (e.g., 2, 10, or e) and a positive number x.
- For antilog, use the "Antilog" tab with a base and logarithm value as input; result = b^y.
- On the "Equation" tab, enter a log equation of the form log_b(ax+c) = d to solve for x.
- Verify logarithm properties on the "Properties" tab by entering two numbers to see proofs of the addition, subtraction, and exponent rules.
- Use the standard log reference table (log 2 ~ 0.301, log 3 ~ 0.477, ln 2 ~ 0.693) for manual validation.
- Watch the domain: x must be > 0 and b > 0 with b != 1, otherwise the result is undefined.
🧮 Logarithm and Change of Base
log_b(x) = y iff b^y = x; log_b(x) = ln(x) / ln(b) = log(x) / log(b)
- b = logarithm base (b > 0 and b != 1)
- x = numerus or argument (x > 0)
- y = logarithm value (real number)
- ln = natural logarithm with base e ~ 2.71828
- log = base-10 logarithm (common logarithm)
Key properties: log(ab) = log(a) + log(b), log(a/b) = log(a) - log(b), log(a^n) = n * log(a).
💡 Worked example: Computing log_2(50)
Given:- Base b = 2
- Numerus x = 50
- Use change-of-base to base 10
Steps:- log_2(50) = log(50) / log(2)
- log(50) = log(5 x 10) = log 5 + log 10 = 0.699 + 1 = 1.699
- log(2) ~ 0.301
- log_2(50) = 1.699 / 0.301 ~ 5.644
- Verification: 2^5.644 = 2^5 x 2^0.644 = 32 x 1.5625 ~ 50
Result: log_2(50) ~ 5.6439, meaning 2 raised to the power 5.6439 gives 50.
❓ Frequently asked questions
What is the difference between log, ln, and log_2?
log without a subscript usually means base-10 (common logarithm), used in science and engineering. ln is the natural logarithm with base e ~ 2.71828, dominant in calculus and probability. log_2 (base 2) is heavily used in computer science (information, entropy). All three are linked by the change-of-base formula.
Why must a logarithm's base not be 1?
If b = 1, then 1^y = 1 for every y, so log_1(x) has no unique solution for x != 1 and has infinitely many solutions for x = 1. Likewise, b must be positive so that b^y is defined for all real y. These rules are standard in introductory algebra.
What is logarithm used for in the real world?
The Richter scale for earthquakes, the decibel scale for sound, and pH for solutions are all logarithmic base 10. Compound interest and radioactive half-life use ln. Binary search and tree data structures use log_2 for O(log n) complexity. Logarithms convert multiplication into addition, very useful for data spanning wide ranges.
How do I compute a logarithm without a calculator?
Memorize the basics: log 2 ~ 0.301, log 3 ~ 0.477, log 7 ~ 0.845. Use log(ab) = log a + log b for decomposition, e.g., log 6 = log 2 + log 3 = 0.778. For other bases, use change-of-base. Four-decimal log tables were standard high-school equipment before calculators became common (pre-1980s).
What is a logarithmic equation and how do I solve it?
A log equation is one whose variable x lies inside a logarithm, e.g., log_2(x+1) = 3. Solve by rewriting in exponential form b^d = ax+c, then x = (b^d - c)/a. For the example: 2^3 = x+1, so x = 7. Always verify the domain conditions x > 0 and numerus > 0.
📚 Sources & references
Last updated: May 11, 2026