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If we have an expression as ${{5}^{x}}={{\left( 0.5 \right)}^{y}}=1000$, then which of the following is true?
A. $x>y$
B. $x< y$
C. $x=y$
D. $y >x$

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Last updated date: 13th Jul 2024
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Answer
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Hint: For solving this question you should know about taking the log of any function. In this problem first we will take the logarithmic term or we will take the log of the equation which is given for x and y values. And then by solving these terms, we will find the values for x and y. And then we will compare the values of both and then we will select the correct option.

Complete step-by-step solution:
According to our question it is asked if ${{5}^{x}}={{\left( 0.5 \right)}^{y}}=1000$, then which of the given options would be correct. As we know that if any number or any term is given in a form as a variable and a number and the equivalent value is also given, then we will solve that by taking a log of that term. And in this process, we will first take the log of the given terms, which will provide us the values of desired variables. And then we will solve these terms and find the values of desired variables. So, if we look at our question, then,
${{5}^{x}}={{\left( 0.5 \right)}^{y}}=1000$
If we want to find the value of x, then we will take only ${{5}^{x}}=1000$.
If we take the log of both sides, then,
$\begin{align}
  & x={{\ln }_{5}}1000 \\
 & x={{\ln }_{5}}{{10}^{3}} \\
 & x=3{{\ln }_{5}}10 \\
\end{align}$
Now, if we find the value of y, then we will select the term as ${{\left( 0.5 \right)}^{y}}=1000$.
If we take the log on both sides, then,
$\begin{align}
  & y={{\ln }_{0.5}}1000 \\
 & y={{\ln }_{0.5}}{{10}^{3}} \\
 & y=3{{\ln }_{0.5}}10 \\
\end{align}$
Now, if we compare the values of x and y, then it is clear that the base of y is less than the base of x. So, the value of $x>y$.
Hence the correct answer is option A.

Note: While solving these types of questions, you have to be careful about taking the log of the terms and solve the calculations very carefully because if there is any calculation mistake, then your entire question would be incorrect.