Logarithms were developed in the 17th century by the Scottish mathematician, John Napier. They were a clever method of reducing long multiplications into much simpler additions (and reducing divisions into subtractions).

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Young John Napier had to help his dad, who was a tax collector.
John got sick of multiplying and dividing large numbers all day and devised logarithms to make his life easier.

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The use of logarithms made trigonometry and many other fields of mathematics much simpler to calculate.

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When calculus was developed later in the century, logarithms became central to many solutions. Today, logarithms are still important in many fields of science and engineering, even though we use calculators for most simple calculations.

Logarithmic Functions :

Logarithmic Functions A logarithm is simply an exponent that is written in a special way.

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For example, we know that the following exponential equation is true:
32 = 9
In this case, the base is 3 and the exponent is 2. We can write this equation in logarithm form (with identical meaning) as follows:
log39 = 2

log39 = 2 :

log39 = 2 We say this as "the logarithm of 9 to the base 3 is 2". What we have effectively done is to move the exponent down on to the main line.
This was done historically to make multiplications and divisions easier, but logarithms are still very handy in mathematics.

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The logarithmic function is defined as:
f(x) = logbx
The base of the logarithm is b. The 2 most common bases that we use are base 10 and base e, which we meet in Logs to base 10 and Natural Logs (base e) in later sections.

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The logarithmic function has many real-life applications, in acoustics, electronics, earthquake analysis and population prediction.

Example 1: Write in logarithm form: 8 = 23 :

Example 1: Write in logarithm form: 8 = 23 Solution: log28 = 3
This just follows from the definition of a logarithm.

Example 2: Write in exponential form: log101000 = 3 :

Example 2: Write in exponential form: log101000 = 3 Solution: 1000 = 103
Once again, this just follows from the definition of a logarithm.

Example 3: Find b if :

Example 3: Find b if SOLUTION:

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