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Most simply a logarithm is a way of answering the question:
How many of one number do we need to get another number. How many of x do we need to get y
More formally:
x raised to what power gives me y
Below is an example of a logarithm:
\log _{3} 9
We read it:
log base 3 of 9
And it means:
3 raised to what power gives me 9?
In this case the answer is easy: 3^2 gives me nine, which is to say: three
multiplied by itself.
Using exponents to calculate logarithms
This approach becomes rapidly difficult when working with larger numbers. It's
not as obvious what \log \_{5} 625 would be using this method. For this
reason, we use exponents which are intimately related to logarithms.
A logarithm can be expressed identically using exponents for example:
\log _{3} 9 = 2 \leftrightarrow 3^2 = 9
By carrying out the conversion in stages, we can work out the answer to the question a logarithm poses.
Let's work out \log \_{2} 8 using this method.
-
First we add a variable (x) to the expression on the right hand:
\log _{2} 8 \leftrightarrow x -
Next we take the base of the logarithm and combine it with x as an exponent. Now our formula looks like this:
\log _{2} 8 \leftrightarrow 2^x -
Next we add an equals and the number that is left from the logarithm (8):
\log _{2} 8 \leftrightarrow 2^x = 8
Then the problem is reduced to: how many times do you need to multiply two by itself to get 8? The answer is 3 : 2 x 2 x 2 or 2 p3. Hence we have the balanced equation:
\log _{2} 8 \leftrightarrow 2^3 = 8
Common base values
Often times a base won't be specified in a log expression. For example:
\log1000
This is just a shorthand and it means that the base value is ten, i.e that the logarithm is written in denary (base 10). So the above actually means:
\log _{10} 1000 = 3
This is referred to as the common logarithm
Another frequent base is Euler's number (approx. 2.71828) known as the natural logarithm
An example
\log _{e} 7.389 = 2