Question

If ${25^{x - 1}} = {5^{2x - 1}} - 100$, then the value of $x$ is _____________.
(A) $3$
(B) $2$
(C) $4$
(D) $1$

Answer 1

Hint: We are given an equation with exponentials and a variable. Check if the variable is on the base or the exponential part of the given term. If the variable is present at the base of the term then we can proceed to solving the equation directly to find the variable, but if the variable is at the exponential part then we need to find a way to equate the exponential parts.

Complete step by step solution:
Let us note down the given linear equation:
$ \Rightarrow {25^{x - 1}} = {5^{2x - 1}} - 100$
Here we see that the variable is part of the exponential part in the equation so we have to rearrange the equation in a way that we will be able to equate the powers to find ‘$x$’.
So we solve this question by the method of ‘equating the powers’.
From the given question we know that $25 = {5^2}$, in a similar way we can change the left hand side that is:
${25^{(x - 1)}}$ can be ${5^{2(x - 1)}}$
$ \Rightarrow {5^{2(x - 1)}} = {5^{2x - 2}}$ is the original equation’s left hand side.
So the original equation becomes:
$ \Rightarrow {5^{2x - 2}} = {5^{2x - 1}} - 100$
Now separate the terms containing the variable ‘$x$’:
$ \Rightarrow {5^{2x - 1}} - {5^{2x - 2}} = 100$
Now the terms on the left side can be expanded as:
$ \Rightarrow ({5^{2x}} \times {5^{ - 1}}) - ({5^{2x}} \times {5^{ - 2}}) = 100$
From the left hand side, we proceed to take out the common term ${5^{2x}}$ :
$ \Rightarrow {5^{2x}} \times ({5^{ - 1}} - {5^{ - 2}}) = 100$
This equation can also be written as:
$ \Rightarrow {5^{2x}} \times (\dfrac{1}{5} - \dfrac{1}{{{5^2}}}) = 100$
Then simplifying we can see:
$ \Rightarrow {5^{2x}} \times (\dfrac{{5 - 1}}{{25}}) = 100$
$ \Rightarrow {5^{2x}} \times (\dfrac{4}{{25}}) = 100$
Isolate the term with the variable:
$ \Rightarrow {5^{2x}} = 100 \times (\dfrac{{25}}{4})$
$ \Rightarrow {5^{2x}} = 25 \times 25$
If the bases are equal we can equate the powers so we try to bring a common base on both sides:
$ \Rightarrow {5^{2x}} = {5^2} \times {5^2}$
$ \Rightarrow {5^{2x}} = {5^4}$
Now we equate the powers since the bases are both $5$ so they are equal.
$ \Rightarrow 2x = 4$
Solving for ‘$x$’ we get:
$ \Rightarrow x = 2$
Now since we have solved the given equation we got the value of $x$ as $x = 2$.

Therefore, the value of $x=2$, and the correct option is option (B).

Note:
There are multiple ways to solve this question; we can either equate the powers as we already did or we can use a variable substitution method. In the substitution method we can give a variable to the common terms present on either side of the given equation. Here the common term would be ${5^{2x}}$, so we could substitute a variable say ‘$y$’ for this common term.