# add log of different bases

Discussion in 'Undergraduate Math' started by darklord, Jan 15, 2005.

1. ### darklordGuest

Hi,

I have a question regarding logarithm. I know how to add and subtract
log of a same base, however I have this problem where I must add two
logs of different bases. Here is the problem, which I am supposed to
find x without using calculator.

log(8)4 - log(8)X = log(8)5 + log(16)6

The logarithm of base 16 throws me off. My guess is that I probably
can break log(16)6 into two log(8) but I don't know whether the 6
would still stay the same? Anyways, could you please show me how to
solve this problem? Thanks

darklord, Jan 15, 2005

2. ### Brian VanPeltGuest

There is a "change of base" formula for logarithms.

log(a) x = ( log(b) x ) / ( log(b) a )

where a and b are positive numbers, not equal to 1. Using this, you
can rewrite log(16) 6 as

log(16) 6 = ( log(8) 6 ) / ( log(8) 16 )

Also, you might notice that 8^(4/3) = 16, so log(8) 16 = 4/3. Thus,

( log(8) 6 ) / ( log(8) 16 ) = ( log(8) 6 ) / ( 4/3 )

= ( 3/4 ) ( log(8) 6 )

= log(8) 6^(3/4)

Hope this helps,

Brian

Brian VanPelt, Jan 15, 2005

3. ### Patrick FitzGeraldGuest

Hint

Look through your Book or notes for the formula for convering logs
from one base to another base, then use that to change base 16 to
base 8

Patrick

Patrick FitzGerald, Jan 15, 2005
4. ### Stan BrownGuest

Log_16(x) = log_8(x) / log_8(16)

Since 8^(4/3) = 16, log_8(16) = 4/3

Therefore log_16(6) = log_8(6) / (4/3) = (3/4) * log_8(6) =
log_8(6^(3/4))

Stan Brown, Jan 15, 2005