Question

How do you solve ${2^x} = 5$ ?

Answer 1

Hint:
So for solving this mathematical expression we will use the concept of logarithmic. Therefore, we will take the logarithm on both sides and take the $x$ one side rest on the other, and then we will be able to solve the above expression.

Formula used:
Logarithmic property is given by,
$\log {a^b} = b\log a$
Here, $a\& b$ will be the variables.

Complete step by step solution:
we have the expression given as ${2^x} = 5$
Now for solving it, we will take the logarithmic both the side, so the equation will be equal to
$ \Rightarrow \log {2^x} = \log 5$
Now by using the logarithmic property and solving the left side of the equation, we will get the expression as
$ \Rightarrow x\log 2 = \log 5$
Now on solving for the value of $x$ , so for this, we will take the rest of the term to the right side of the expression and it will be in the denominator, so we will get the expression as
$ \Rightarrow x = \dfrac{{\log 5}}{{\log 2}}$
As we know that the value of $\log 5$ is $0.6989$ and the value of $\log 2$ will be $0.3010$ , so on substituting the values, we will get the expression as
$ \Rightarrow x = \dfrac{{0.6989}}{{0.3010}}$
And on dividing the numerator by the denominator, we will get the expression as
$ \Rightarrow x \approx 2.32$

Therefore, on simplifying ${2^x} = 5$, we get $2.32$.

Note:
While solving the problem based on the logarithmic. We should keep in mind that one will be some basic properties and the other one will be the values of the logarithmic up to some extent. If we will memorize it then it will always help in solving the questions based on it. And to divide such numbers easily we should always consider the approximation.