Question

How do you solve ${3^{x - 1}} = 27$ ?

Answer 1

Hint: To solve the given equation, firstly we need to make the bases the same on both sides. Then by equating the exponents we can find the value of the variable. We will use that if the bases are the same then we can equate the exponents.

Complete step-by-step solution:
The given equation is ${3^{x - 1}} = 27$
We see that in the given equation the variable lies in the exponent of the term. So, here to find the solution or to find the value of the variable, we need to convert the right side in the form of the exponent $3$ .
This way we can equate the exponents and evaluate the value of the variable. So this becomes
$ \Rightarrow {3^{x - 1}} = {3^3}$
Now, as the bases on both sides are equal, then the exponents are also equal to each other.
$ \Rightarrow x - 1 = 3$
Adding $1$ on both sides,
$ \Rightarrow x - 1 + 1 = 3 + 1$ , as positive and negative of the same term becomes zero, so
$ \Rightarrow x = 4$ .

$x = 4$ is the solution of ${3^{x - 1}} = 27$.

Note: The expression ${a^m}$ is read as $a$ raised to the power $m$ . Here, $m$ is the exponent and the number $a$ is the base. The exponent denotes the number of times the base is to be multiplied. There are also other laws of exponents that are used to solve the equations.
In the topic, laws of exponents, there is a law that means in an equation if bases on both the sides are the same and the exponents contain the algebraic expression, then to evaluate the value of the variable, we can equate the exponents.