The definition of the logarithm: it is the inverse of exponentiation. If, for x, a and b are positive reals,
a^x = b
Then the base a logarithm of b is x,
log_a b = log_a a^x = x log_a a = x
Change of Base
The first question is to consider the importance of the base of the logarithm. To some extent, the base is unimportant. Logarithms of different bases are related by scale factors. That is, comparing a logarithm to base a to a logarithm to base be is like comparing feet to inches: there is some multiplier which converts between them.
The formula that I prefer to remember is:
log_a b log_b c = log_a c
I find this formulat easy to remember because it looks
like you are cancelling the b's, similar to fractions:
(b/a) (c/b) = (c/a)
Proof:
Use both sides as the exponent of "a" and see that they
end up equal. On the left,
a^(log_a b log_b c) = (a^(log_a b))^(log_b c)
= b^(log_b c)
= c.
On the right,
a^(log_a c) = c.( There is a bit more to this proof, since it depends on the fact that for positive a if,
a^x = a^ythen we can correctly conclude that,
x = y.Anyone who has seen an exponential curve "knows" that it increases steadily, so that this property is true.)
Other formulas
From this formula, one can derive that:
log_a b log_b a = log_a a = 1So that, for instance,
log_7 8 = 1/(log_8 7)Another interesting property of logs is,
x^(log_a y) = (a^(log_a x))^(log_a y)
= a^(log_a x log_a y)
= (a^(log_a y))^(log_a x)
= y^(log_a x)
So that, for instance,
7^(log_2 8) = 8^(log_2 7)