# Logarithm Calculator

This calculator can be used to determine any type of logarithm of a real number of any base you wish. Common, binary and natural logarithms can all be found using the online logarithm calculator.

## Definition of a Logarithm

A logarithm of a real number is the exponent to which a base, that is, a different fixed number, needs to be increased in order to generate that real number.

If x = ay, then y = loga x

Where:

a, x, y are real numbers, x > 0, a > 0, a ≠ 1, and a is the base of the logarithm.

To illustrate, take the number 10,000 to base 10. The logarithm of this real number will be 4. This is because 10,000 is equivalent to 10 to the power of 4. Thus, just as division is the opposite mathematical operation to multiplication, the logarithm is the opposite operation to exponentiation.

Traditionally, a base of 10 is assumed in logarithms, but a base can be any number (except 1). The binary logarithm of x is typically written as log2 x or lb x. However, a base of e is typically written as ln x and rarely as loge x.

As illustrated above, logarithms can have a variety of bases. A binary logarithm, or a logarithm to base 2, is applied in computing, while the field of economics utilizes base e, and in education base 10, written simply as log x, log10 x or lg x, is used. By organizing numbers according to these bases, real numbers can be expressed far more simply.

Log Calculator

log

Base a Name for logax ISO 31-11 notation Notes
10 Common logarithm lg x lg x = log10 x
e Natural logarithm ln x ln x = loge x
2 Binary logarithm lb x lb x = log2 x

## Logarithm Rules

1. Zero Rule: loga 1 = 0

2. Identity Rule: loga a = 1

3. Log of Power Rule: loga a x = x

4. Power of Log Rule: a loga x = x

5. Product Rule: loga (xy) = loga x + loga y

6. Quotient Rule: loga (x/y) = loga x - loga y

7. Power Rule: loga xn = n loga x

8. Change of Base Rule: loga x = logb x × loga b

9. Base Switch Rule: logb a = 1 / loga b

10. Change of Base Rule: logb x = loga x / loga b

Where: x > 0, y > 0, a > 0, b > 0 ; a ≠ 1, b ≠ 1 ; n is any real number.

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