Question

Logarithm Question

Find x, if \(\cfrac{6}{5} a^{\log_a x \log_{10} a \log_a 5 } - 3^{\log_{10} (x/10)} = 9^{\log_{100} x + \log_4 {2}} \)

Answer 1

\(\LARGE{\cfrac{6}{5} a^{\log_a x \log_{10} a \log_a 5 } - 3^{\log_{10} (x/10)} = 9^{\log_{100} x + \log_4 {2}}}\)

Answer 2

@hartnn @ganeshie8

Answer 3

those are lots of logs :P

Answer 4

Haha :D

Answer 5

Should I simplify RHS first?

Answer 6

easiest simplification i see is for log_4 2

Answer 7

Haha.. :D

But, RHS becomes :

\(\LARGE 3^{\log_{10} x + 1}\)

Answer 8

maybe use this repeatedly for LHS : mlogn = logn^m

Answer 9

left side , 1st term exponent!
simplifies greatly

Answer 10

Yeah..

Answer 11

doing it..

\(\LARGE \cfrac{6}{5} x^{\log_{10} a \log_a {5}}\)

Answer 12

the first term in LHS ^

Answer 13

\(\LARGE \cfrac{6}{5} x^{\cfrac{1}{\log _ a{10} } \times \log_a 5}\)

hmm, no:/

Answer 14

Well, I will transpose that second term to RHS :

\(\LARGE \cfrac{6}{5} x^{\log_{10} a \log_a {5}} = 3^{\log_{10} (x/10) } + 3^{\log_{10} x + 1}\)

@hartnn
from where did that come from? o.O

Answer 15

\(\large a^{\log_a x \log_{10} a \log_a 5 } \text{ simplifies to } 5^{\log_{10} x}\)

Answer 16

use change of base repeatedly

Answer 17

or shall I do like this :

\(\LARGE \cfrac{6}{5} x^{\cfrac{\log_x a}{\log_x {10}} \log_a 5}\)

\(\LARGE \cfrac{6}{5} a^{\cfrac{\log_a 5}{\log_x {10}}}\)

\(\LARGE \cfrac{6}{5} 5^ {\cfrac{1}{\log_x {10}}}\)

Answer 18

or \(\LARGE \cfrac{6}{5} 5^{\log_{10} x}\)

o.O finally got a short expression.

Answer 19

Okay, so , it becomes :

\(\LARGE \cfrac{6}{5} 5^{\log_{10} x} = 3^{\log_{10} (x/10)} + 3^{\log_{10} x + 1}\)

Answer 20

\(\LARGE 6 \times 5^{\log_{10} (x-1) } = 3^{\log_{10} x } \times \left(\cfrac{1}{3} + 3\right) \)

Answer 21

I think, we got it :

if I keep it 6/5 then :

\(\LARGE {\cfrac{6}{5} \times 5^{\log_{10} x } = 3^{\log_{10} x } \times \cfrac{10}{3} \\

\left(\cfrac{3}{5} \right)^2 = \left(\cfrac{3}{5}\right) ^{\log_{10} x}} \)



\(\boxed{\bf{x = 100}} \ddot \smile\)

Answer 22

Got it.. Thanks @ganeshie8 and @hartnn .

Answer 23

Wow ! lol im still wasting my paper... looks great :)

Answer 24

Haha...! Purchase some more for me... more to come. :)