Question

Find the value of \[\dfrac{1}{x} - \dfrac{1}{y}\] if \[{\left( {2.3} \right)^x} = {\left( {0.23} \right)^y} = 1000\]

Answer 1

Hint: Use the logarithmic functions by taking two cases so that the exponential terms comes down to bases and then solve each case to get the value of x and y respectively to put it in \[\dfrac{1}{x} - \dfrac{1}{y}\].

Complete step-by-step answer:
It is given that \[{\left( {2.3} \right)^x} = {\left( {0.23} \right)^y} = 1000\]
We will pair each one of them with 1000 to get different answers
Let us take the first case as \[{\left( {2.3} \right)^x} = 1000\]
Now this can be written as
 \[{\left( {2.3} \right)^x} = {\left( {10} \right)^3}\]
Taking log in both the sides we get it as
\[\begin{array}{l}
\therefore \log {\left( {2.3} \right)^x} = \log {\left( {10} \right)^3}\\
 \Rightarrow x\log \left( {2.3} \right) = 3\log 10\\
 \Rightarrow x\log \left( {2.3} \right) = 3\\
 \Rightarrow x = \dfrac{3}{{\log (2.3)}}\\
 \Rightarrow \dfrac{1}{x} = \dfrac{{\log (2.3)}}{3}
\end{array}\]
As we have the value of \[\dfrac{1}{x}\] let us try to find the value of \[\dfrac{1}{y}\]
We have our second case as \[{\left( {0.23} \right)^y} = 1000\]
This can also be written as
\[{\left( {0.23} \right)^y} = {\left( {10} \right)^3}\]
Taking log in both the sides we get it as
\[\begin{array}{l}
\therefore \log {\left( {0.23} \right)^y} = \log {\left( {10} \right)^3}\\
 \Rightarrow y\log \left( {0.23} \right) = 3\log 10\\
 \Rightarrow y\log \left( {0.23} \right) = 3\\
 \Rightarrow y = \dfrac{3}{{\log (0.23)}}\\
 \Rightarrow \dfrac{1}{y} = \dfrac{{\log (0.23)}}{3}
\end{array}\]
As we also have the value of \[\dfrac{1}{y}\]
Let us put it in \[\dfrac{1}{x} - \dfrac{1}{y}\]
\[\begin{array}{l}
\therefore \dfrac{{\log (2.3)}}{3} - \dfrac{{\log (0.23)}}{3}\\
 = \dfrac{{\log (2.3) - \log (0.23)}}{3}\\
 = \dfrac{{\log \left( {\dfrac{{2.3}}{{0.23}}} \right)}}{3}\\
 = \dfrac{{\log 10}}{3}\\
 = \dfrac{1}{3}
\end{array}\]
Therefore \[\dfrac{1}{3}\] is the correct answer.
Note: The logarithms that have been used in this question are all \[{\log _{10}}\]. Also we could have just took the value of and x and y individually and then solve \[\dfrac{1}{x} - \dfrac{1}{y}\] by taking LCM but that will be log method and the chances of making mistakes were also high so better taking the value of \[\dfrac{1}{x}\& \dfrac{1}{y}\] individually and then just subtract them.