Question

If we are given an equation as \[{5^{x - 2}} = 5\], then find the value of \[x\].

Answer 1

Hint: According to the question, exponent of \[5\] on the left-hand side is \[x - 2\]. As we know, if power is not mentioned in term it will be taken as \[1\]. So, the exponent of \[5\] which is at the right-hand side, will be \[1\]. By equating power on both sides, we can get the value of \[x\].

Complete step-by-step solution:
Let us consider the given equation to be an exponential equation. As given in the question, the two expressions have the same base.
We know that if the two expressions are equal, then their exponents must also be equal. This is because, if you have \[2\] to the power \[a\], and \[2\] to the power \[b\], but \[a\] is not equal to \[b\] then, \[2\] to the power \[a\] cannot have the same value as \[2\] to the power \[b\].
So, here in the given equation \[{5^{x - 2}} = 5\Rightarrow x - 2 = 1\]( as bases are equal therefore exponents will be equal).
\[ \Rightarrow x = 2 + 1\]
\[ \Rightarrow x = 3\]
Therefore, the value of \[x = 3\] which satisfies the given expression.

Note: This type of expression generally comes under the category of exponential expression which does not require logarithm for solving. This is also a topic included under algebra where we can solve the equation by making their bases the same. So, it is also known as solving exponential equations with the same base.