Welcome to **Logarithm**.app. On this site you can find everything about logarithms.

Every inverse function f^{−1}(x) is a function that undoes another function f(x).

It essentially means that if an input x of the function f(x) produces y as output, then the input y of the inverse function f^{−1}(x) produces the output x. And vice versa!

Logarithm are the inverse functions to exponentiations.

log

## Logarithm Definition

**b ^{y} = x ⇔ y = log_{b}(x)**

b, x, y ∈ R^{+}, b ≠ 1

R^{+} denotes the set of positive real numbers.

### Logarithm Parts

**b** is the **base**

**x** is the **argument**

**y** is the **exponent**

## Logarithm Examples

Logarithmic Form | Exponential form | |
---|---|---|

log_{2}(8) = 3 | ⇔ | 2^{3} = 8 |

log_{10}(1000) = 3 | ⇔ | 10^{3} = 1000 |

ln(x) = 1 | ⇔ | e^{0} = 1 |

log_{3}(9) = 2 | ⇔ | 3^{3} = 9 |

log(100) = 2 | ⇔ | 10^{2} = 100 |

ln(x) is the **natural logarithm** which uses the number **e** as base. You may think of it as log_{e}(x).

The logarithm with base 2 is called **binary logarithm**, and the logarithm with base 10 is usually called **decimal logarithm**.

The decimal logarithm, which is also known as **common logarithm** and **decadic logarithm**, often has it’s base omitted:

**log _{10}(x) = log(x)**

The graph below depicts ln(x), log_{2}(x) as well as log(x) for small values of x.

## Evaluating Logarithms

1. Assumed you want to solve this equation:

log_{10}(1000000) = y.

Write it in the equivalent form:

10^{y} = 1000000

Take the log with base 10 from both sides:

log_{10}10^{y} = log_{10}(1000000)

y = 6

2. Supposed you want to solve:

log_{2}(32) = y.

Write it in the equivalent form:

2^{y} = 32

Take the log with base 2 from both sides:

log_{2}2^{y} = log_{2}(32)

y = 5

Practice makes experts, so you may soon find yourself skipping the procedure and evaluating a log just by asking, “b to what power is y”?

3. Try solving log_{4}(64) by asking yourself “4 to what power is 64?”