Question

How do you solve the algebraic expression ${5^{2x}} = 8$?

Answer 1

This problem deals with finding the solution of $x$, with the help and applications of logarithms to the given exponential equation. Here some basic and fundamental identities or properties of logarithms are used in order to solve the given exponential equation such as:
$\Rightarrow \log {a^n} = n\log a$

Complete step-by-step solution:
Given an expression in exponential form where the exponent is varying in the variable $x$.
Consider the given exponential equation below as shown:
$\Rightarrow {5^{2x}} = 8$
Now applying natural logarithms on both sides of the given above equation, as shown below:
$\Rightarrow {\log _e}{5^{2x}} = {\log _e}8$
Here we know that the basic identity or the property of logarithms which is $\log {a^n} = n\log a$, applying this property to the left hand side of the above equation, as shown below:
$\Rightarrow 2x{\log _e}5 = {\log _e}8$
Now divide the above equation by ${\log _e}5$, on both sides of the above equation, as shown below:
$\Rightarrow 2x = \dfrac{{{{\log }_e}8}}{{{{\log }_e}5}}$
Now simplifying the right hand side of the above equation, by substituting the values of ${\log _e}8 = 2.079$ and ${\log _e}5 = 1.609$, in the above equation as shown below:
$\Rightarrow 2x = \dfrac{{2.079}}{{1.609}}$
Simplifying the above expression on the right hand side of the equation and then dividing the equation by 2, to get the value of $x$, as given below:
$\Rightarrow 2x = 1.292$
$\therefore x = 0.646$
The solution of ${5^{2x}} = 8$ is equal to 0.646.

Note: Please note that the above problem is solved with the help of logarithms, that is by applying the logarithms on both sides of the equation to get the value of the variable $x$. Likewise there are other similar properties of logarithms which can be used as applications in order to solve such kind of problems such as:
$\Rightarrow \log ab = \log a + \log b$
$\Rightarrow \log \left( {\dfrac{a}{b}} \right) = \log a - \log b$
$\Rightarrow {e^{{{\log }_e}a}} = a$