94 lines
2.1 KiB
Markdown
94 lines
2.1 KiB
Markdown
---
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categories:
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- Mathematics
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tags: [algebra]
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---
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Most simply a logarithm is a way of answering the question:
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>
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> How many of one number do we need to get another number. How many of x do we need to get y
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More formally:
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>
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> x raised to what power gives me y
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Below is an example of a logarithm:
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$$ \log \_{3} 9
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$$
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We read it:
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>
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> log base 3 of 9
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And it means:
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>
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> 3 raised to what power gives me 9?
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In this case the answer is easy: $3^2$ gives me nine, which is to say: three multiplied by itself.
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## Using exponents to calculate logarithms
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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.
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A logarithm can be expressed identically using exponents for example:
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$$ \log \_{3} 9 = 2 \leftrightarrow 3^2 = 9
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$$
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By carrying out the conversion in stages, we can work out the answer to the question a logarithm poses.
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Let's work out $\log \_{2} 8$ using this method.
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1. First we add a variable (x) to the expression on the right hand:
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$$ \log \_{2} 8 \leftrightarrow x
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$$
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1. Next we take the base of the logarithm and combine it with x as an exponent. Now our formula looks like this:
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$$ \log \_{2} 8 \leftrightarrow 2^x
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$$
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1. Next we add an equals and the number that is left from the logarithm (8):
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$$ \log \_{2} 8 \leftrightarrow 2^x = 8
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$$
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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:
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$$ \log \_{2} 8 \leftrightarrow 2^3 = 8
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$$
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## Common base values
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Often times a base won't be specified in a log expression. For example:
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$$ \log1000
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$$
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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:
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$$ \log \_{10} 1000 = 3
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$$
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This is referred to as the **common logarithm**
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Another frequent base is Euler's number (approx. 2.71828) known as the **natural logarithm**
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An example
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$$ \log \_{e} 7.389 = 2
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$$
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