Question

PLEASE HELP ME!

(Logarithm Question) Prove that:

Answer 1

\[\frac{ 1 }{ \log _{x}10 } + \frac{ 1 }{ \log _{y}10} = \frac{ 1 }{ \log _{x _{y}}10}\]

Answer 2

add on the left, then use the change of base formula, if i remember correctly
seems like this question comes up every month or so

Answer 3

hi @satellite73 !! since you've helped me so much before, when you are done here could you possibly help me with my recent question?

Answer 4

I`m sorry but I am totally clueless on this question :S Can you show me how to do it plz

Answer 5

this is going to be a bear to write
add on the left you get
\[\frac{\log_x(10)+\log_y(10)}{\log_x(10)\times \log_y(10)}\]now change of base. doesn't matter what base you use, so you might as well use base ten
\[\frac{\frac{\log(10)}{\log_x(10)}+\frac{\log(10)}{\log_y(10)}}{\frac{\log(10)}{\log_x(10)}\times \frac{\log(10)}{\log_y(10)}}\]

Answer 6

now since \(\log(10)=1\) you can rewrite all this mess as
\[\frac{\frac{1}{\log_x(10)}+\frac{1}{\log_y(10)}}{\frac{1}{\log_x(10)}\times \frac{1}{\log_y(10)}}\]

Answer 7

now add in the numerator, get
\[\frac{\frac{\log_y(10)+\log_x(10)}{\log_x(10)\log_y(10)}}{\frac{1}{\log_x(10)\log_y(10)}}\]

Answer 8

denominators cancel gives
\[\log_x(10)+\log_y(10)\]

Answer 9

would that be the final answerÉ

Answer 10

no, we are not quite done
we have to show the right hand side is the same thing

Answer 11

all that was on the left.
the right hand side is
\[\frac{1}{\log_{xy}(10)}\] and again by change of base this is
\[\frac{1}{\frac{\log(10)}{\log(xy)}}=\log(xy)=\log(x)+\log(y)\] oh damn i messed up somewhere

Answer 12

ok it is much easier that i made it out to be
the right hand side is
\[\log(x)+\log(y)\] by what i wrote directly above, the change of base
now lets do the same on the left

Answer 13

\[\frac{1}{\log_x(10)}=\frac{1}{\frac{\log(10)}{\log(x)}}=\frac{1}{\frac{1}{\log(x)}}=\log(x)\]
similarly
\[\frac{1}{\log_y(10)}=\log(y)\]

Answer 14

so both the left hand side and the right hand side turn in to
\[\log(x)+\log(y)\]and i hope it is clear that when i write \(\log(x)\) i mean \(\log_{10}(x)\)

Answer 15

yea its making sense

Answer 16

ok it wasn't as hard as i made it out to be
just the change of base formula on both sides to rewrite them both in base 10

Answer 17

Thank you!

Answer 18

Thanks a lot for the help!