Question

solve this .
Required nyc answer ☺

Answer 1

According to my visibility in Question :
Q : Given that
$log_{2}(3) = a \: and \: \: log_{3}(5) = b \\ \: and \: log_{7}(2) = c \: \: \: then \\ \: express \: the \: number \: log_{140}(63) .$
For Calculation see pic

Final Answer :
$\frac{2ac}{2c + abc + 1}$

See Pic 1 : Identies which are used in Calculation
See Pic 2 ,3: Calculation of given expression

Understanding required :
1) Base changing theorem in logarithm
2) Basic Logarithmic Manipulations.

Answer 2

Logical Analysis :

1) First we will write all parameters in terms of log :
$a = log_{2}(3) \\ b = log_{3}(5) \\ c = log_{7}(2) \\ \\ ac = log_{7}(3) \\ ab = log_{2}(5) \\ abc = log_{7}(5)$



2) Now, we have to find in terms of, a,b,c
$log_{140}(63)$
We know 140 , 63 both are divisible by 7 , so we will change it in base 7 for easy manipulation.



3)
$log_{140}(63) = \frac{ log_{7}(63) }{ log_{7}(140) } \\ = > \frac{ log_{7}(7) + log_{7}(9) }{ log_{7}(7) + log_{7}(20) } \\ = > \frac{1 + 2 log_{7}(3) }{1 + log_{7}(5) + 2 log_{7}(2) } \\ = > \frac{1 +2ac }{1 + abc + 2c}$