Logarithms Overview

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.

Meaning of 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 loga x pronounced log of x to base a or the base-a, logarithm of x or the logarithm, base a of x.

Types of logarithms

There are two types of Logarithms:

  1. Common logarithm: A logarithm with base 10. For example `log_(10) 5`, `log_(10) 7`, `log 7`, `log_(10) C`
  2. 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.

Rules of Logarithms

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

  1. `log_a a = 1`
  2. `log_a 1 = 0`
  3. `log_a X^n = nlog_a X`
  4. `log_a X + log_a Y = log_a (X*Y)`
  5. `log_a X - log_a Y = log_a (X/Y)`
  6. `log_a X = (log X)/(log a)`
  7. `log_a X = log_b X * log_a b`
  8. `log_a b * log_b a = 1`
  9. `log_a b = 1/(log_b a)`
  10. `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.

Meaning of 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 loga x pronounced log of x to base a or the base-a, logarithm of x or the logarithm, base a of x.

Types of logarithms

There are two types of Logarithms:

  1. Common logarithm: A logarithm with base 10. For example `log_(10) 5`, `log_(10) 7`, `log 7`, `log_(10) C`
  2. 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.

Rules of Logarithms

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

  1. `log_a a = 1`
  2. `log_a 1 = 0`
  3. `log_a X^n = nlog_a X`
  4. `log_a X + log_a Y = log_a (X*Y)`
  5. `log_a X - log_a Y = log_a (X/Y)`
  6. `log_a X = (log X)/(log a)`
  7. `log_a X = log_b X * log_a b`
  8. `log_a b * log_b a = 1`
  9. `log_a b = 1/(log_b a)`
  10. `a^(log_a X) = X`

Now, let's move in. Solved Logarithmic questions

Recommended Lessons