Question 419359
If you're talking about
{{{log(2, (19))}}}
then this is called "the base 2 logarithm of 19". It represents the exponent you would put on a 2 to get a 19:
{{{2^log(2, (19)) = 19}}}
I think your problem is to find out what number is this exponent you put on a 2 to get 19.<br>
Some logarithms are easy to find. For example {{{log(5, (26))}}} is the exponent for 5 that results in 25. We know what exponent on a 5 results in 25. It is 2. So {{{log(5, (25)) = 2}}}.<br>
But what exponent on a 2 results in 19? {{{2^4 = 16}}} and {{{2^5 = 32}}} so the exponent that results in 18 is going to be somewhere between 4 and 5. This is about as far as we can reasonably go without the help of a calculator.<br>
But how do we use a calculator to find {{{log(2, (19))}}}? There is no button for base 2 logarithms. The base of a logarithm can be any positive number (except 1). So it would seem calculators would need an infinite number of buttons to handle all the different possible logarithms.. Instead of having a button for each logarithm your calculator your calculator has buttons for logarithms of just one or two bases, 10 and/or "e". The base 10 log button has "log" on it and the base e log button has "ln" on it. To find logarithms of other bases you use the change of base formula:
{{{log(a, (p)) = log(b, (p))/log(b, (a)){{{
to express the logarithm you're looking for in terms of base 10 or base e logarithms. Then you can use your calculator. Changing your logarithm your logarithm into base 10 logarithms we get:
{{{log(2, (19)) = log((19))/log((2))}}}
Or we could use base e logarithms:
{{{log(2, (19)) = ln(19)/ln(2)}}}
Both of these will work and they will both result in the same answer. Just use your calculator to find the two logarithms and then divide them.