**b ^{c} = a, b must be greater than 1 then c = log_{b} (a)**

We say that c is the logarithm to the base b of a. Here are two examples which are familiar from arithmetic.

**10 ^{2} = 100; 2 = log_{10} (100) and 5^{3} = 125; 3 = log_{5} (125)**

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**Rules of loges**

- Log
_{a}(x, y) = Log_{a}(x) + Log_{a}(y); - Log
_{a}(x/y) = Log_{a}(x) – Log_{a}(y); - Log
_{a}(x^{n}) = n log_{a}(x)

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**Logs to base 10 and logs to base e **

Logs to base 10 are used in science and engineering to express either very large numbers or very small numbers and are normally written as “log” without a 10 subscript, they are called “logs” or “common logs”. Large numbers have positive powers e.g. 10^{23} and very small numbers have negative powers e.g. 10^{-34}.

Logs to base e are normally written as “In” rather than loge and they are called “natural logs” or “In”. They are important for their Maths properties which model many natural phenomena, hence they appear very often in science and engineering. The natural number e comes from the series below.

**e ^{x} = 1 + x + x^{2}/2! + x^{3}/3! + x^{4}/4! …. ….**

The symbol “…” means “continues on for ever” and is called an “ellipsis”. The symbol ‘1″ means the factorial of the preceding number. For the special case when x = 1 we have

**e = 1 + 1 + x ^{2}/2! + x^{3}/3! + x^{4}/4! …. …= 2.7182819…..**

Note that e is a number and it is fixed, it is not a variable. This is similar to 71 or any other number e.g. 10 or 3.6, so it is not written in italic but upright roman script. In science and engineering we quite often find equations with e^{-x} in them, this is just a special case of ex which alternates the sign of the terms for even and odd power of x.

**e ^{-x} = 1 – x + x^{2}/2! – x^{3}/3! + x^{4}/4! …. ….**

y = e^{x} may also be written as y = exp(x) with “exp” all in lower case. This exp notation may be used for greater clarity particularly in printed material. An example from Chemistry is the Arrhenius equation k = Ae^{-Ea/RT} which can also be written as k = A exp(—E_{a} / RT). Be careful in using calculators as they sometimes have a key marked EXP, all capitals, which means “multiply by 10 raised to the power”, which you then enter the power into the calculator.

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**Additional Roles:**

**Log _{a }(a) = 1;**

**Log _{a }(a^{x}) = x;**

**Log _{a }(b) = 1 / Log_{b }(a)**