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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 q^{y} = x.

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

Two equalities (q^{y} = x et y = log_{q}(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 log
_{3}(81):

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

3×3×3×3 = 81 ⇨ 3^{4} = 81 hence log_{3}(81) = 4 : the base 3 logarithm of 81 is 4.

- Calculate log
_{2}(64):

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

2×2×2×2×2×2 = 64 ⇨ 2^{6} = 64 hence log_{2}(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 log
_{100}(1)?

To what power must we raise 100 to obtain 1?

100^{y} = 1 ⇨ 100^{0} = 1 ⇨ log_{100}(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. (log_{q}(1) = 0).

- Analyse the base q logarithmic function
- Understand reciprocal relationship between the exponential and logarithmic functions.
- Study the influence of the parameters A and k in the general expression f(x) = A·log
_{q}(kx). Illustrate the properties of logarithms.

x = q^{y} can also be written in the exponential form: x = exp_{q}(y).

The two relations x = exp_{q}(y) et y = log_{q}(x) are therefore equivalent.

- The exponential is the result of the power: x is…