Answer: y = 4.6051701859881
loge(100) = 4.6051701859881
e4.6051701859881 = 100
Logarithm Function Curve
Complete Logarithm Calculator Guide & Information
1. What is a Logarithm?
A logarithm is the inverse operation of exponentiation. It answers the question: "To what power must we raise the base to obtain the given number?" If by = x, then logb(x) = y. The logarithm of x with base b is the exponent y to which b must be raised to produce x.
Logarithms were invented by John Napier in the early 17th century as a way to simplify complex calculations. Before electronic calculators, logarithms turned multiplication into addition and division into subtraction, dramatically speeding up hand computation.
2. Basic Definition
logb(x) = y if and only if by = x
where b > 0, b ≠ 1, and x > 0
3. Common Logarithm Bases
| Base |
Name |
Notation |
Common Uses |
| e ≈ 2.71828 |
Natural logarithm |
ln(x) or loge(x) |
Calculus, physics, engineering, growth/decay |
| 10 |
Common logarithm |
log(x) or log10(x) |
Science, engineering, decibels, pH scale |
| 2 |
Binary logarithm |
log2(x) or lb(x) |
Computer science, information theory, music |
4. Logarithm Rules and Identities
| Rule |
Formula |
Example |
| Product Rule |
logb(xy) = logb(x) + logb(y) |
log2(4×8) = log2(4) + log2(8) = 2 + 3 = 5 |
| Quotient Rule |
logb(x/y) = logb(x) − logb(y) |
log10(1000/10) = 3 − 1 = 2 |
| Power Rule |
logb(xn) = n × logb(x) |
log2(8³) = 3 × log2(8) = 3 × 3 = 9 |
| Change of Base |
logb(x) = logc(x) / logc(b) |
log2(10) = ln(10) / ln(2) ≈ 3.3219 |
| Reciprocal Rule |
logb(1/x) = −logb(x) |
log10(0.01) = −log10(100) = −2 |
| Identity |
logb(b) = 1 |
log5(5) = 1 |
| Log of 1 |
logb(1) = 0 |
log10(1) = 0 |
| Inverse Property |
blogb(x) = x |
10log10(100) = 100 |
5. Change of Base Formula
The change of base formula allows calculating logarithms with any base using a calculator that only has ln or log buttons. This is the formula used by this calculator internally.
logb(x) = ln(x) / ln(b) = log10(x) / log10(b)
6. Properties of the Logarithm Function
- Domain: x > 0 (only positive numbers have real logarithms)
- Range: All real numbers (−∞ to +∞)
- X-intercept: Always at (1, 0) because logb(1) = 0 for any valid base
- Monotonicity: Strictly increasing if b > 1; strictly decreasing if 0 < b < 1
- Asymptote: Vertical asymptote at x = 0 (function approaches −∞ as x → 0+)
- Continuity: Continuous and differentiable everywhere on its domain
- Derivative: d/dx ln(x) = 1/x
- Integral: ∫ ln(x) dx = x·ln(x) − x + C
7. Real-World Applications
- Decibel scale: Sound intensity measured in dB uses base-10 logarithmic scale
- Richter scale: Earthquake magnitude is a base-10 logarithmic measure
- pH scale: Acidity/alkalinity defined as −log10[H⁺]
- Information theory: Binary logarithm measures information content in bits
- Finance: Calculating compound interest, doubling time, and return rates
- Radioactive decay: Half-life calculations use natural logarithms
- Signal processing: Frequency response plots (Bode plots) use log scales
- Statistics: Log-normal distributions and log transformations
- Algorithm analysis: Many algorithms have O(log n) time complexity
8. Input & Control Definitions
- Argument (top input): The number you want to take the logarithm of. Must be positive (greater than 0). You can enter any positive number or the letter 'e' for Euler's number.
- Base (bottom input): The base of the logarithm. Must be positive and not equal to 1. Common values: e (natural log), 10 (common log), 2 (binary log).
- Result (right input): The computed logarithm value y such that basey = argument.
- Calculate Button: Computes the logarithm and displays the result with verification.
- Clear Button: Clears all input fields completely.
9. Worked Examples
Example 1 — Natural log: Calculate ln(100)
- loge(100) = ln(100) ≈ 4.60517
- Verification: e4.60517 ≈ 100 ✓
Example 2 — Base-10 log: Calculate log10(1000)
- log10(1000) = 3
- Verification: 10³ = 1000 ✓
Example 3 — Binary log: Calculate log2(256)
- log2(256) = 8
- Verification: 2⁸ = 256 ✓
10. Important Notes
- Logarithm of zero or negative numbers is undefined in the real number system (results in complex numbers).
- Base must be positive and cannot equal 1.
- log(0) approaches negative infinity.
- For very large arguments, results may lose floating-point precision.
- Natural logarithm (base e) is the most commonly used logarithm in calculus and higher mathematics.
11. Related Mathematical Concepts
- Exponentiation: The inverse operation of taking a logarithm
- Natural exponential function: ex, inverse of natural logarithm
- Antilogarithm: Raising the base to the power of the logarithm result
- Double logarithm: log(log(x)), appears in algorithm analysis
- Polylogarithm: Generalization of the logarithm function
- Discrete logarithm: Logarithm in modular arithmetic, basis for some cryptosystems
12. References
1. Napier, John. "Mirifici Logarithmorum Canonis Descriptio." 1614.
2. Euler, Leonhard. "Introduction to Analysis of the Infinite." 1748.
3. Stewart, James. "Calculus: Early Transcendentals." Cengage Learning. 2015.
4. Apostol, Tom M. "Calculus, Volume 1." Wiley. 1991.
5. Knuth, Donald E. "The Art of Computer Programming, Volume 1." Addison-Wesley. 1997.
6. Shannon, Claude E. "A Mathematical Theory of Communication." 1948.