Understanding Logarithms
A logarithm allows us to reverse the operation of exponentiation. While exponents allow us to grow numbers rapidly (e.g., 2³ = 8), logarithms tell us "how many times" we need to multiply a base to reach a certain number.
The fundamental definition is:
logb(x) = y
is equivalent to
by = x
Key Components
- bBase: The number being multiplied (must be Positive and ≠ 1).
- xArgument (Value): The result we are analyzing (must be Positive).
- yLogarithm: The exponent required to get x from b.
Logarithm Rules & Formulas
These properties are essential for simplifying complex logarithmic expressions.
| Rule Name | Formula | Example |
|---|---|---|
| Product Rule | log(xy) = log(x) + log(y) | log(6) = log(2) + log(3) |
| Quotient Rule | log(x/y) = log(x) - log(y) | log(2) = log(6) - log(3) |
| Power Rule | log(xᵖ) = p · log(x) | log(x²) = 2 · log(x) |
| Change of Base | log_b(x) = ln(x) / ln(b) | log₂(8) = ln(8)/ln(2) |
| Zero Rule | log_b(1) = 0 | log₁₀(1) = 0 |
| Identity Rule | log_b(b) = 1 | log₅(5) = 1 |
Common Logarithm Values (2026)
The table below shows common logarithm values for base 10 (log₁₀) and natural logarithm (ln) calculations. These values are frequently used in scientific, engineering, and mathematical applications.
| Value (x) | log₁₀(x) | ln(x) | log₂(x) | Application |
|---|---|---|---|---|
| 1 | 0.000 | 0.000 | 0.000 | Base case |
| 2 | 0.301 | 0.693 | 1.000 | Binary systems |
| 10 | 1.000 | 2.303 | 3.322 | Decimal base |
| e (2.718) | 0.434 | 1.000 | 1.443 | Natural base |
| 100 | 2.000 | 4.605 | 6.644 | Century, percentage |
| 1000 | 3.000 | 6.908 | 9.966 | Kilo- prefix |
| 0.1 | -1.000 | -2.303 | -3.322 | Decimals |
| 0.5 | -0.301 | -0.693 | -1.000 | Half-life |
Values rounded to 3 decimal places. Use Change of Base Formula: log_b(x) = ln(x) / ln(b) for custom bases.
Real-World Applications
pH Scale (Chemistry)
The acidity of a solution is determined by the negative base-10 logarithm of the hydrogen ion concentration: pH = -log₁₀[H+].
Decibels (Sound)
Sound intensity is measured in decibels (dB), which uses a logarithmic scale. An increase of 10 dB represents a 10-fold increase in sound intensity.
Richter Scale
Earthquake magnitude is logarithmic. A magnitude 6.0 earthquake has 10 times the shaking amplitude of a magnitude 5.0 earthquake.
Logarithms are closely related to exponents and powers. When working with exponential functions, you may need to calculate powers and roots as part of solving logarithmic equations.
Frequently Asked Questions
- What is a logarithm?
- A logarithm is the inverse operation of exponentiation. If b^y = x, then log_b(x) = y. It answers the question: 'To what power must I raise the base (b) to get the value (x)?' For example, since 10^2 = 100, log₁₀(100) = 2.
- What is the natural logarithm (ln)?
- The natural logarithm (ln) uses the mathematical constant e (approximately 2.71828) as its base. It is written as ln(x) = log_e(x). Natural logarithms are crucial in calculus, physics, and modeling exponential growth or decay.
- What is the difference between log and ln?
- 'log' usually refers to the common logarithm with base 10 (log₁₀), while 'ln' refers to the natural logarithm with base e. However, in advanced mathematics and computer science contexts, 'log' might sometimes imply base e or base 2, so it's always good to check the context.
- How do I calculate a logarithm with a custom base?
- You can calculate a logarithm with any base 'b' using the Change of Base Formula: log_b(x) = ln(x) / ln(b) or log_b(x) = log₁₀(x) / log₁₀(b). This is how most calculators compute custom bases.