# Thread: logarithm

1. ## logarithm

$\displaystyle \ln (r^ { 2 } s^ { 8 } \sqrt[ 8 ]{ r^ { 5 } s^ { 2 } } )$ is equal to $\displaystyle A \ln r + B \ln s$

find A & B

im not really sure what to do on this problem, but split up stuff inside
$\displaystyle \ln (r^ { 2 }) + \ln (s^ { 8 }) + \ln(\sqrt[ 8 ]{ r^ { 5 })} + \ln(\sqrt[8]{ s^ { 2 } } )$

$\displaystyle 2\ln r + 8\ln s + \frac{1}{8}(r^5) + \frac{1}{8}(s^2)$

a lil stuck at this part

2. Originally Posted by viet
$\displaystyle \ln (r^ { 2 } s^ { 8 } \sqrt[ 8 ]{ r^ { 5 } s^ { 2 } } )$ is equal to $\displaystyle A \ln r + B \ln s$

find A & B

im not really sure what to do on this problem, but split up stuff inside
$\displaystyle \ln (r^ { 2 }) + \ln (s^ { 8 }) + \ln(\sqrt[ 8 ]{ r^ { 5 })} + \ln(\sqrt[8]{ s^ { 2 } } )$

$\displaystyle 2\ln r + 8\ln s + \frac{1}{8}(r^5) + \frac{1}{8}(s^2)$

a lil stuck at this part
Whoah, not so fast there. don't split everything up so quick, simplify first.

$\displaystyle \ln (r^2 s^8 \sqrt [8]{r^5 s^2}) = \ln (r^2 s^8 (r^5 s^2)^{ \frac {1}{8}})$
....................$\displaystyle = \ln (r^2 s^8 r^{ \frac {5}{8}} s^{ \frac {1}{4}})$
....................$\displaystyle = \ln (r^{ \frac {21}{8}} s^{ \frac {33}{4}})$

i think you can take it from here

3. Originally Posted by viet
$\displaystyle \ln (r^ { 2 } s^ { 8 } \sqrt[ 8 ]{ r^ { 5 } s^ { 2 } } )$ is equal to $\displaystyle A \ln r + B \ln s$

find A & B

im not really sure what to do on this problem, but split up stuff inside
$\displaystyle \ln (r^ { 2 }) + \ln (s^ { 8 }) + \ln(\sqrt[ 8 ]{ r^ { 5 })} + \ln(\sqrt[8]{ s^ { 2 } } )$

$\displaystyle 2\ln r + 8\ln s + \frac{1}{8}(r^5) + \frac{1}{8}(s^2)$

a lil stuck at this part
Hello, Viet,

you nearly missed the correct solution...

I take your last line and add the missing parts:

$\displaystyle 2\ln r+8\ln s+\frac{1}{8}\ln {(r^5)} + \frac{1}{8}\ln {(s^2)} = 2\ln r+8\ln s+\frac{5}{8}\ln {(r)} + \frac{2}{8}\ln {(s)} =$
$\displaystyle \left(2 + \frac{5}{8} \right)\ln(r) + \left(8 + \frac{1}{4} \right)\ln(s)$

I'll leave the rest for you