Question

How do you solve ${11^{x - 1}} = 1331$?

Answer 1

Hint: To solve this question, we need to make a linear equation with the given variable. However, here the term including variable is given as the power of 11. Therefore, we need to make the right hand side of the equation with some power of 11 so that after that by equating the powers, we can solve for the variable $x$.

Complete step by step answer:
We are given the equation ${11^{x - 1}} = 1331$.
We will now convert the right hand side of the equation as a power of 11. For that we will first divide 1331 by 11.
$ \Rightarrow \dfrac{{1331}}{{11}} = 121$
Therefore, we can rewrite 1331 as:
$1331 = 121 \times 11$
Now we will divide 121 by 11.
$ \Rightarrow \dfrac{{121}}{{11}} = 11$
Hence, 121 cab be rewritten as:
$121 = 11 \times 11$
Thus, 1331 can be written as:
$1331 = 11 \times 11 \times 11 = {11^3}$
Therefore, our main equation becomes
${11^{x - 1}} = {11^3}$
As we can see that both sides of the equations has the same base 11. Therefore, its power can be equated with each other.
$ \Rightarrow x - 1 = 3$.
Now, this has become the linear equation with variable $x$ which we can solve easily.
We will add 1 to both the sides of the equation to get our final answer.
$
   \Rightarrow x - 1 + 1 = 3 + 1 \\
   \Rightarrow x = 4 \\
 $
Hence, our final answer is 4.

Note: In this question, after converting the right side of the equation with power of 11, we have equated powers of both the sides to get the linear equation. We can also apply the concept of logarithms to solve this equation as:
${11^{x - 1}} = {11^3}$
Let us take logarithms on both the sides,
$ \Rightarrow \log {11^{x - 1}} = \log {11^3}$
Now, we will apply the exponent rule for logarithms.
$ \Rightarrow \left( {x - 1} \right)\log 11 = 3\log 11$
As we can see, $\log 11$will get cancelled out.
$ \Rightarrow x - 1 = 3$
Now, we got the same linear equation by solving which we can get our final answer as 4.