SOLUTION: Logarithms using change of base formula; log2(19) The 2 is very small and on the bottom and the 19 is large. Can you help me to understand how to do this problem? Thanks Debbie

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Logarithms using change of base formula; log2(19) The 2 is very small and on the bottom and the 19 is large. Can you help me to understand how to do this problem? Thanks Debbie      Log On


   



Question 419359: Logarithms using change of base formula;
log2(19)
The 2 is very small and on the bottom and the 19 is large.
Can you help me to understand how to do this problem? Thanks
Debbie

Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
If you're talking about
log%282%2C+%2819%29%29
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%5Elog%282%2C+%2819%29%29+=+19
I think your problem is to find out what number is this exponent you put on a 2 to get 19.

Some logarithms are easy to find. For example log%285%2C+%2826%29%29 is the exponent for 5 that results in 25. We know what exponent on a 5 results in 25. It is 2. So log%285%2C+%2825%29%29+=+2.

But what exponent on a 2 results in 19? 2%5E4+=+16 and 2%5E5+=+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.

But how do we use a calculator to find log%282%2C+%2819%29%29? 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:

Or we could use base e logarithms:
log%282%2C+%2819%29%29+=+ln%2819%29%2Fln%282%29
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.