Question

How do you solve ${5^x} = {25^{x - 1}}$?

Answer 1

This problem deals with solving the expressions with exponents and bases. An expression that represents repeated multiplication of the same factor is called a power. The number 5 is called the base, and the number $x$ is called the exponent, on the left hand side of the given equation. The exponent corresponds to the number of times the base is used as a factor.
Here some basic rule of exponents and bases are used here such as:
$\Rightarrow {\left( {{a^m}} \right)^n} = {a^{mn}}$
$\Rightarrow {a^m} = {a^n}$, if the bases are the same, then the exponents have to be the same.

Complete step-by-step answer:
The given equation is ${5^x} = {25^{x - 1}}$, which is considered below:
$\Rightarrow {5^x} = {25^{x - 1}}$
Now consider the right hand side of the equation which is ${25^{x - 1}}$, as shown below:
Here the base is 25, which is a perfect square, which is the square of the number 5, as shown below:
$\because 25 = {5^2}$
Substituting this in the right hand side of the expression which is ${25^{x - 1}}$, as shown below:
$\Rightarrow {25^{x - 1}} = {\left( {{5^2}} \right)^{x - 1}}$
Now applying the rule that ${\left( {{a^m}} \right)^n}$ is equal to ${a^{mn}}$, as shown below:
$\Rightarrow {25^{x - 1}} = {5^{2\left( {x - 1} \right)}}$
Now equating the left hand side and the right hand side of the given equation, as shown below:
$\Rightarrow {5^x} = {25^{x - 1}}$
But we obtained that ${25^{x - 1}} = {5^{2\left( {x - 1} \right)}}$,
$\Rightarrow {5^x} = {5^{2\left( {x - 1} \right)}}$
Here the exponents on both sides have to be equal, as shown:
$\Rightarrow x = 2\left( {x - 1} \right)$
$\Rightarrow x = 2x - 2$
Now rearranging the above equation such that:
$\Rightarrow 2 = 2x - x$
$\therefore x = 2$

Note:
Please note that usually a power is represented with a base and an exponent. The base tells what number is being multiplied. The exponent, a small number written above and to the right of the base number, tells how many times the base number is being multiplied. The product rule says that to multiply two exponents with the same base, you keep the base and add the powers.
$\Rightarrow {a^m} \cdot {a^n} = {a^{m + n}}$