The reason this is so is based on an understanding of what logarithms are. The idea behind logarithms is that it is possible to take any positive number, raise it to the correct power and get any other positive number as a result. For example, it is possible to raise 7 to some power and get a 12.
Some of these exponents are easy to figure out. For example:
What exponent for 2 results in 4? Answer: 2
What exponent for 3 results in 9? Answer: 2
What exponent for 81 results in 9? Answer: 1/2 (Remember 1/2 as an exponent means square root!)
What exponent for 13 results in 1? Answer: 0 (Remember any non-zero number to the zero power is a 1!)
What exponent for 8 results in 1/8? Answer: -1 (Remember an exponent of -1 means reciprocal!
Logarithms are used to express these exponents. In general expresses the exponent for a that results in b. So, using the examples above:
Many of these exponents are hard to know. For example, what power of 14 results in 22? Without knowing exactly what this exponent is we can still express it as:
So why is ?
Well, the exponent, , stands for the exponent for 3 that results in 7. And where to we find ? Answer: As the exponent on a 3! So if we raise 3 to the power for 3 the results in 7, we will of course get a 7!
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