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Logarithms
Before continuing, it is important that we make sure we have an understanding of what a logarithm is: the reason is that many measurements and values in sound engineering are expressed in a logarithmic scale, we will deal with the first of them in the next lesson.

A logarithm is defined as:

"a quantity representing the power to which a fixed number (the base) must be raised to produce a given number" [1]

In other words we can say that logarithm are the inverse operation of a root extraction.

In the function

cuberoot



x is the number that raised to the third is equal to 8, which is 2. By extracting roots, we find a base (x) to a given exponent (3), in order for it to be equal to a given value (8). So we answer to the question 'what is the number that raised to the third is equal to 8?'

But what if we want to find to what power (x) should a given number (a) be raised in order for it to be equal to another given number (b)?

power



this is when logarithms come into place:

if:

2_to_x



then:

logtwo16



'2' is called the base of the logarithm

logtwosol



The most common base in sound engineering is 10.

For example:

log10_1



log10_2



log10_3



if you see a logarithmic function with no base indicated, that means that the base is 10:

log1000_1



when writing a function of a log base 10 of a given number, the 10 is understood, so

log1000_2



A few things to remember:

rem1_a because rem1_b




rem2_a because rem2_b




rem3_a because rem3_b




if rem4_a then rem4_b






Laws of Logarithms:

1. log product is equal to the sum of logs.

law1_a



Example:

law1_b



2. log of a quotient is equal to the difference between logs.

law2_a



Example:

law2_b



3. log number to a power equals power times log of a number. (we are going to use this soon!)

law3



base change:

basechange



where a and b are any valid base



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[1] New Oxford American Dictionary, 2005