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Logarithmic function

To what power must we raise a number(q) to obtain another number (x)?

The theologist, physician, astronomist and mathematician, John Napier introduced the concept of logarithms to resolve this question.

The problem consists of isolating the variable y in the equation qy = x.

The solution is written y = logq(x) and can expressed as: y is the base q logarithm of x.

Two equalities (qy = x et y = logq(x)) express a similar relation between three numbers: x (the result of the power), y (the exponent) and q (the number raised the power, called the base).

The logarithm of base 10 is the decadic logarithm, written log(x).

The logarithm of base e is the natural logarithm, written ln(x).

Examples:

  • Calculate log3(81): 

To what power must we raise the number 3 to obtain 81? 

3×3×3×3 = 81 ⇨ 34 = 81 hence log3(81) = 4 : the base 3 logarithm of 81 is 4.

  • Calculate log2(64): 

To what power must we raise the number 2 to obtain 64? 

2×2×2×2×2×2 = 64 ⇨ 26 = 64 hence log2(64) = 6 : the base 2 logarithm of 64 is 6.

  • log(100) = 2: 

We must square the number 10 to obtain 100 (here the logarithm is decadic)

  • and log100(1)? 

To what power must we raise 100 to obtain 1?

100y = 1 ⇨ 1000 = 1 ⇨ log100(1) = 0 : the base 100 logarithm of 1 is 0.

Any number raised to the power of 0 is 1. The logarithm of 1 is always 0 no matter what the base is. (logq(1) = 0).