Question

# If $a = \log {}_23,$ $b = \log {}_25$ and $c = \log {}_72$, then $\log {}_{140}63$ in terms of a ,b ,c is a) $\dfrac{{2a + 1}}{{2a + abc + 1}}$b) $\dfrac{{2ac + 1}}{{2a + c + a}}$c) $\dfrac{{2ac + 1}}{{2c + ab + a}}$d) None of these

Hint: Use logarithmic rules to solve this problem.
1.$\log {}_ba = \dfrac{{\log a}}{{\log b}}$
2.$\log (a \times b) = \log a + \log b$
3.$\log {a^x} = x\log a$

Given that, $a = \log {}_23,$ $b = \log {}_25$ and $c = \log {}_72$
We will use the first rule from the hint. Then
$a = \dfrac{{\log 3}}{{\log 2}}$, $b = \dfrac{{\log 5}}{{\log 2}}$ and $c = \dfrac{{\log 2}}{{\log 7}}$ .
Now, moving towards the value we have to find
$\log {}_{140}63$
=$\dfrac{{\log 63}}{{\log 140}}$ Using $\log {}_ba = \dfrac{{\log a}}{{\log b}}$
=$\dfrac{{\log (9 \times 7)}}{{\log (2 \times 70)}}$ here factors should be used according to data given.
= $\dfrac{{\log 9 + \log 7}}{{\log 2 + \log 70}}$ using $\log (a \times b) = \log a + \log b$
=$\dfrac{{\log 9 + \log 7}}{{\log 2 + \log (2 \times 5 \times 7)}}$ Factorize number 70.
=$\dfrac{{\log {3^2} + \log 7}}{{\log 2 + \log 2 + \log 5 + \log 7}}$ 9 can be written as square of 3 .
=$\dfrac{{2\log 3 + \log 7}}{{2\log 2 + \log 5 + \log 7}}$ using $\log {a^x} = x\log a$
=$\dfrac{{2a\log 2 + \dfrac{{\log 2}}{c}}}{{2\log 2 + b\log 2 + \dfrac{{\log 2}}{c}}}$ $\log 3 = a\log 2,\log 5 = b\log 2,\log 7 = \dfrac{{\log 2}}{c}$
rearranging the log terms so that all come in log2 form
=$\dfrac{{\dfrac{{c \times 2a\log 2 + \log 2}}{c}}}{{\dfrac{{c \times 2\log 2 + bc \times \log 2 + \log 2}}{c}}}$ taking LCM separately for numerator and denominator.
=$\dfrac{{\log 2(2ac + 1)}}{{\log 2(2c + bc + 1)}}$ cancelling C and taking log2 common.
=$\dfrac{{2ac + 1}}{{2c + bc + 1}}$
Thus,
$\log {}_{140}63$=$\dfrac{{2ac + 1}}{{2c + bc + 1}}$
Option d is the correct answer.

Note: When we solve problems related to logarithm we should try to simplify the ratios as much as we can.
If there is a need to split a number ,we should factorize it using the numbers in the given question.[like 63 and 140]. It is to make your calculations easy.
Don’t miss even a single step like finding L.C.M. or taking common terms.