Calculate logarithms with any base including natural log (ln), common log (log₁₀), binary log (log₂), and custom bases. Solve logarithmic equations and find antilog values with step-by-step solutions.
A logarithm is the inverse operation of exponentiation. When we write log_b(x) = y, we're asking "to what power must we raise base b to get x?" The answer is y, meaning b^y = x. Logarithms are fundamental to mathematics, science, and engineering, with applications ranging from measuring earthquake intensity to calculating compound interest.
Understanding logarithms is essential for advanced mathematics, as they simplify complex calculations involving exponential growth and decay, solve equations with unknown exponents, and provide powerful tools for data analysis and scientific measurement.
Where: b = base (b > 0, b ≠ 1), x = argument (x > 0), y = logarithm result
| Expression | Base | Value | Result | Verification |
|---|---|---|---|---|
| log₁₀(100) | 10 | 100 | 2 | 10² = 100 |
| ln(e) | e | e ≈ 2.718 | 1 | e¹ = e |
| log₂(8) | 2 | 8 | 3 | 2³ = 8 |
| log₁₀(1) | 10 | 1 | 0 | 10⁰ = 1 |
| log₅(125) | 5 | 125 | 3 | 5³ = 125 |
| ln(1) | e | 1 | 0 | e⁰ = 1 |
| log₂(0.5) | 2 | 0.5 | -1 | 2⁻¹ = 0.5 |
Scientific Measurements: Many scientific scales use logarithms to handle large ranges of values. The Richter scale for earthquakes, pH scale for acidity, and decibel scale for sound intensity are all logarithmic.
Compound Interest and Finance: Logarithms help calculate how long it takes for investments to reach certain values under compound interest. The rule of 72 and various financial formulas rely on logarithmic calculations.
Population Growth: Exponential population growth models use logarithms to determine doubling times and to solve for unknown time periods or growth rates.
Computer Science: Logarithms are fundamental in algorithm analysis (time complexity), information theory (bits and entropy), and data structures (balanced trees).
Identity Rules:
• log_b(1) = 0 (any base raised to power 0 equals 1)
• log_b(b) = 1 (any base raised to power 1 equals itself)
• log_b(b^x) = x (logarithm and exponentiation cancel out)
• b^(log_b(x)) = x (exponentiation and logarithm cancel out)
Negative and Fractional Arguments:
• log_b(1/x) = -log_b(x) (reciprocal becomes negative logarithm)
• log_b(√x) = ½log_b(x) (square root becomes half the logarithm)
• log_b(x^(1/n)) = (1/n)log_b(x) (nth root becomes 1/n times logarithm)
Euler's Number e: The natural logarithm uses base e ≈ 2.71828, a mathematical constant that appears naturally in calculus, probability, and many areas of mathematics.
Calculus Connection: The derivative of ln(x) is 1/x, making natural logarithms particularly useful in calculus and differential equations.
Continuous Compounding: Natural logarithms model continuous exponential growth and decay processes, including radioactive decay and population dynamics.
Type 1 - Single Logarithm: log_b(x) = c → x = b^c
Example: log₂(x) = 5 → x = 2⁵ = 32
Type 2 - Multiple Logarithms: Use logarithm properties to combine or separate terms before solving.
Example: log(x) + log(x-3) = 1 → log(x(x-3)) = 1 → x(x-3) = 10¹ → x² - 3x - 10 = 0
Type 3 - Exponential Form: Convert to exponential form when logarithms appear with different bases.
Example: log₂(x) = log₃(27) → x = 2^(log₃(27)) = 2³ = 8
Domain Errors: Remember that logarithms are only defined for positive arguments. log(0) and log(negative numbers) are undefined in real numbers.
Property Misapplication: log(x + y) ≠ log(x) + log(y). The sum rule only applies to products: log(xy) = log(x) + log(y).
Base Confusion: Always specify the base clearly. ln means base e, log typically means base 10, but context matters in different fields.
Exponent Distribution: (log(x))² ≠ log(x²). Powers of logarithms are different from logarithms of powers.
Logarithmic Differentiation: Used in calculus to differentiate complex functions by taking the natural logarithm first, then differentiating.
Logarithmic Integration: The integral of 1/x is ln|x| + C, making logarithms essential for integration techniques.
Complex Logarithms: In complex analysis, logarithms can be extended to complex numbers, leading to multiple-valued functions and branch cuts.