Log Calculator
A log calculator finds the logarithm of a number for a specified base. Logarithms are used in mathematics, algebra, science, engineering, computer science, finance, and many calculations involving exponential relationships.
What Is a Logarithm?
A logarithm tells you the exponent needed to raise a given base to produce a number. In other words, if:
bx = y
then:
logb(y) = x
For example:
23 = 8
Therefore:
log2(8) = 3
Common Logarithm Bases
The most commonly used logarithms are base 10 and base e.
A common logarithm uses base 10 and is written as:
log10(x)
A natural logarithm uses the mathematical constant e β 2.71828 and is written as:
ln(x)
Logarithms can also use other positive bases, such as 2, 3, or 5.
How to Calculate a Logarithm
Suppose you want to find log2(32).
Ask which power of 2 equals 32:
25 = 32
Therefore:
log2(32) = 5
Common Log Examples
| Expression | Result |
|---|---|
| log10(10) | 1 |
| log10(100) | 2 |
| log10(1000) | 3 |
| log2(8) | 3 |
| log2(16) | 4 |
| log5(25) | 2 |
Change of Base Formula
When a calculator does not directly support the desired base, the logarithm can be converted using the change of base formula:
logb(x) = log(x) Γ· log(b)
Natural logarithms can also be used:
logb(x) = ln(x) Γ· ln(b)
For example:
log2(10) = ln(10) Γ· ln(2) β 3.3219
Logarithm Rules
Several important rules make logarithmic calculations easier:
Product rule: logb(xy) = logb(x) + logb(y)
Quotient rule: logb(x/y) = logb(x) β logb(y)
Power rule: logb(xn) = n logb(x)
Valid Inputs for Logarithms
For a real-valued logarithm, the number inside the logarithm must be positive. A logarithm of zero or a negative number is not defined as a real number.
The base must also be positive and cannot equal 1. Therefore, a valid logarithm has a base greater than 0 and different from 1.
Where Are Logarithms Used?
Logarithms are used to solve exponential equations and describe quantities that change over very large or very small ranges. Applications include sound levels, acidity, scientific scales, compound growth, algorithms, and signal analysis.
Common Mistakes to Avoid
Make sure you use the correct base. A common logarithm uses base 10, while a natural logarithm uses base e. Also check that the input value is positive when working with real logarithms.
Quick Summary
A logarithm determines the exponent required to produce a number from a particular base. The basic relationship is logb(x) = y when by = x. A log calculator can quickly evaluate logarithms for different valid bases.