Question

How do you solve $ {3^{x - 1}} = 81 $ ?

Answer 1

Hint: In this question, we are given an equation that is in exponential form. When a number is raised to some power, the number is said to be in exponential form, for example $ {a^x} $ is an exponential function. On the left-hand side of the equation, 3 is raised to the power x-1, and the right-hand side contains a constant term, that is, we have to find the value of x from the given relation, thus we have to solve either of the sides to convert it into a form similar to the other side of the equation. Then on comparing the two sides of the equation we get the value of x.

Complete step-by-step answer:
Prime factorization of 81 is done as –
 $
  81 = 3 \times 3 \times 3 \times 3 \\
  81 = {3^4} \;
  $
Comparing the right-hand side with the left-hand side, we get –
 $
  {3^{x - 1}} = {3^4} \\
   \Rightarrow x - 1 = 4 \\
   \Rightarrow x = 5 \;
  $
Hence, when $ {3^{x - 1}} = 81 $ , we get $ x = 5 $
So, the correct answer is “ $ x = 5 $ ”.

Note: When a number is multiplied with itself “n” times, the number is raised to the power “n”, for example, let the number be “a” that is multiplied with itself “n” times, it can be written as $ {a^n} $ . On the prime factorization of 81, we see that it is equal to 3 multiplied with itself 4 times so it is written as $ {3^4} $ . Now the base of both the left-hand side and the right-hand side is the same so their powers can be compared with each other. We can solve similar questions using the same approach that is used in the above-mentioned solution.