Logarithms were invented independently by John Napier, a Scotsman and by Joost Burgi, a Swiss in early 1620s.

Subjects > Mathematics > Logarithms

Before moving direct into the problems of logarithms, let's see what we need to know about logarithms.

"Logarithm" comes from two Greek words; *"logos"* meaning "proportion or ratio" and *"arithmos"* meaning a
"number" which together makes "ratio-number".

**Logarithm** is the inverse function to exponentation, as it answers the question *how many of one number do we
multiply to get another number?*, that means the logarithm of a given number x is the exponent to which another
fixed number, the base x, must be raised to produce that number x.

The logarithms of a positive real number x with respect to base b (a positive real number and b ≠ 1) is the
exponent by which b must be raised to give x.

From above definition, it means: `b^y = X` which implies that `y = log_b X`

log is denoted **log _{a} x** pronounced

There are two types of Logarithms:

**Common logarithm**: A logarithm with base 10. For example `log_(10) 5`, `log_(10) 7`, `log 7`, `log_(10) C`**Natural logarithm**: A logarithm with base `e`, e is a constant whose value is approximately `2.718`. For example `log_e 4 = ln 4`, `log_e x = ln x`,**ln**is used to specify that it is a natural logarithm.

If `X > 0, Y > 0, a > 0, b > 0` and `a ne 1, b ne 1` and `n` is any real number, then;

- `log_a a = 1`
- `log_a 1 = 0`
- `log_a X^n = nlog_a X`
- `log_a X + log_a Y = log_a (X*Y)`
- `log_a X - log_a Y = log_a (X/Y)`
- `log_a X = (log X)/(log a)`
- `log_a X = log_b X * log_a b`
- `log_a b * log_b a = 1`
- `log_a b = 1/(log_b a)`
- `a^(log_a X) = X`

Now, let's move in. **Solved Logarithmic questions**

**Recommended Lessons**

Before moving direct into the problems of logarithms, let's see what we need to know about logarithms.

"Logarithm" comes from two Greek words; *"logos"* meaning "proportion or ratio" and *"arithmos"* meaning a
"number" which together makes "ratio-number".

**Logarithm** is the inverse function to exponentation, as it answers the question *how many of one number do we
multiply to get another number?*, that means the logarithm of a given number x is the exponent to which another
fixed number, the base x, must be raised to produce that number x.

From above definition, it means: `b^y = X` which implies that `y = log_b X`

log is denoted **log _{a} x** pronounced

There are two types of Logarithms:

**Common logarithm**: A logarithm with base 10. For example `log_(10) 5`, `log_(10) 7`, `log 7`, `log_(10) C`**Natural logarithm**: A logarithm with base `e`, e is a constant whose value is approximately `2.718`. For example `log_e 4 = ln 4`, `log_e x = ln x`,**ln**is used to specify that it is a natural logarithm.

If `X > 0, Y > 0, a > 0, b > 0` and `a ne 1, b ne 1` and `n` is any real number, then;

- `log_a a = 1`
- `log_a 1 = 0`
- `log_a X^n = nlog_a X`
- `log_a X + log_a Y = log_a (X*Y)`
- `log_a X - log_a Y = log_a (X/Y)`
- `log_a X = (log X)/(log a)`
- `log_a X = log_b X * log_a b`
- `log_a b * log_b a = 1`
- `log_a b = 1/(log_b a)`
- `a^(log_a X) = X`

Now, let's move in. **Solved Logarithmic questions**

**Recommended Lessons**