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How do you solve \[{{5}^{2x}}=20\]?

Last updated date: 22nd Feb 2024
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IVSAT 2024
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Hint: This question involves an equation having a variable x as an exponent. So, we will use the concept of logarithms to solve for x in \[{{5}^{2x}}=20\] we will start by taking log on both sides by applying the logarithm property \[{{a}^{b}}=b\log a\] and \[{{\log }_{x}}x=1\]. First we have to know that by simple basic math simplifications and operations like addition, multiplication we cannot find the solution to these kinds of problems. We have been given \[{{5}^{2x}}=20\] in the question and we have to solve for x.
\[\Rightarrow {{5}^{2x}}=20\]

Complete step-by-step solution:
Firstly we have to apply log on both sides that is the left and right hand sides of the expression. After the application of the log on both sides and also basic property of \[{{a}^{b}}=b\log a\] we get,
\[\Rightarrow {{\log }_{5}}\left( {{5}^{2x}} \right)={{\log }_{5}}20\]
\[\Rightarrow 2x{{\log }_{5}}5={{\log }_{5}}20\]
We know that by the basic logarithm application that log of any number or variable to the base of the same then it will be one. That is \[{{\log }_{x}}x=1\] so by using this basic property of logarithm we get the expression reduced as follows:
\[\Rightarrow 2x={{\log }_{5}}20\]
Here the reduced equation can be further reduced into the equation by using basic arithmetic logic operation by sending the 2 which is in the left hand side to the right hand side so that it will come to the denominator of the right hand side so it will be further reduced as follows…
\[\Rightarrow x=\dfrac{{{\log }_{5}}20}{2}\]
This equation can be further deduced into a numerical value by knowing the value of the logarithm. Here we use the property of logarithms which is \[{{\log }_{x}}x=1\] and \[{{\log }_{y}}{{x}^{2}}=2{{\log }_{y}}x\].
\[\Rightarrow \dfrac{{{\log }_{5}}\left( 5\times 4 \right)}{2}\]
Here we use the basic property of logarithms that is \[{{\log }_{n}}xy={{\log }_{n}}x+{{\log }_{n}}y\] then equation will be reduced as follows
\[\Rightarrow \dfrac{{{\log }_{5}}5+{{\log }_{5}}4}{2}\]
Here we use the property of logarithm which is \[{{\log }_{x}}x=1\] then the equation will be reduced as follows
\[\Rightarrow x=\dfrac{1+2{{\log }_{5}}2}{2}\]
\[\approx 0.930677\]

Note: We must be careful while performing the calculations and a person who is solving the problems of these type must be having a bit knowledge of the logarithms and must be knowing the basic formula and technique of logarithms of \[{{\log }_{x}}x=1\] and so. We must not make any mistakes in solving problems like if we don’t apply the property of \[{{\log }_{x}}x=1\]we can't do further reduction in the calculation of variables.