Question
Question: How do you solve \[{{5}^{2x}}=20\]?...
How do you solve 52x=20?
Solution
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 52x=20 we will start by taking log on both sides by applying the logarithm property ab=bloga and logxx=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 52x=20 in the question and we have to solve for x.
⇒52x=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 ab=bloga we get,
⇒log5(52x)=log520
⇒2xlog55=log520
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 logxx=1 so by using this basic property of logarithm we get the expression reduced as follows:
⇒2x=log520
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…
⇒x=2log520
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 logxx=1 and logyx2=2logyx.
⇒2log5(5×4)
Here we use the basic property of logarithms that is lognxy=lognx+logny then equation will be reduced as follows
⇒2log55+log54
Here we use the property of logarithm which is logxx=1 then the equation will be reduced as follows
⇒x=21+2log52
≈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 logxx=1 and so. We must not make any mistakes in solving problems like if we don’t apply the property of logxx=1we can't do further reduction in the calculation of variables.