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Question: Prove that \(\dfrac{{{{\log }_a}x}}{{{{\log }_{ab}}x}} = 1 + {\log _a}b\)...

Prove that log⁡axlog⁡abx=1+log⁡ab\dfrac{{{{\log }_a}x}}{{{{\log }_{ab}}x}} = 1 + {\log _a}b

Explanation

Solution

In case of changing the bases, there can be any number at the position of base in case of logarithmic except 1. Any real number different to 1 is possible.
Logarithmic of a number x in the base a and in the base b are related as follows:
log⁡bx=log⁡axlog⁡ab{\log _b}x = \dfrac{{{{\log }_a}x}}{{{{\log }_a}b}}
If we know the values of logarithms in base a, we can find the values of logarithms in base b.
Logarithmic functions can be solved by using properties;
log⁡ba=log⁡alog⁡b{\log _b}a = \dfrac{{\log a}}{{\log b}}
log⁡ab=log⁡a+log⁡b\log ab = \log a + \log b

Complete step-by-step answer:
taking Left hand side
L.H.S = log⁡axlog⁡abx=log⁡xlog⁡alog⁡xlog⁡ab\dfrac{{{{\log }_a}x}}{{{{\log }_{ab}}x}} = \dfrac{{\dfrac{{\log x}}{{\log a}}}}{{\dfrac{{\log x}}{{\log ab}}}}
Simplifying the fraction;
= log⁡xlog⁡x×log⁡ablog⁡a\dfrac{{\log x}}{{\log x}} \times \dfrac{{\log ab}}{{\log a}}
= 1×[log⁡a+log⁡blog⁡a]1 \times \left[ {\dfrac{{\log a + \log b}}{{\log a}}} \right]
Separating the denominator;
= log⁡alog⁡a+log⁡blog⁡a\dfrac{{\log a}}{{\log a}} + \dfrac{{\log b}}{{\log a}}
=1+log⁡blog⁡a= 1 + \dfrac{{\log b}}{{\log a}}
Using base changing property;
=1+log⁡ab= 1 + {\log _a}b= R.H.S.
Hence, L.H.S. = R.H.S.

Note: In the above equation, two of the logarithmic properties have been used i.e.
log⁡ab=log⁡a+log⁡b\log ab = \log a + \log b and
log⁡ba=log⁡alog⁡b{\log _b}a = \dfrac{{\log a}}{{\log b}} but there are many more properties of logarithm i.e. log⁡ab=log⁡a−log⁡b\log \dfrac{a}{b} = \log a - \log b
Mainly, the properties of sum of logarithm, multiplication of the logarithmic function, divisions of logarithmic functions are used. In this equation, logarithm with the different bases is used.
There are two types of logarithmic functions
Common log
Natural log
Common log is having 10 as the base in the logarithmic function. It is represented as
log⁡x=∣! n ‾A∣! n ‾10\log x = \dfrac{{\left| \\!{\underline {\, n \,}} \right. A}}{{\left| \\!{\underline {\, n \,}} \right. 10}}.
Natural log is having ‘e’ at the base of the logarithmic function. It is represented as
x=log⁡aAx = {\log _a}A which is also read as ;
“ x is equal to the logarithmic of A at the base a “
So log⁡aA=x{\log _a}A = x or ax=A{a^x} = A
The most commonly used properties of logarithmic functions are :
log⁡mn=nlog⁡m\log {m^n} = n\log m
log⁡mn=logm+log⁡n\log mn = log m + \log n
log⁡mn=log⁡m−log⁡n\log \dfrac{m}{n} = \log m - \log n
log⁡nm=log⁡mlog⁡n{\log _n}m = \dfrac{{\log m}}{{\log n}}