Question

The equation \[{3^{x - 1}} + {5^{x - 1}} = 34\] has how many solutions.
A. one solution
B. two solution
C. three solution
D. four solution

Answer 1

Hint: We will first consider the right hand side of the expression given and then rewrite the term 34 as $9 + 25$ and after this we will write 9 and 25 in the form of powers and express it in terms of 3 and 5, after this we will compare the powers on both sides to determine the value for $x$.

Complete step by step answer:
We will consider the given expression, \[{3^{x - 1}} + {5^{x - 1}} = 34\]
Now, we will transform the right-hand side of the expression by reducing 34 in terms of 9 and 25,
Thus, we get,
\[
   \Rightarrow {3^{x - 1}} + {5^{x - 1}} = 34 \\
   \Rightarrow {3^{x - 1}} + {5^{x - 1}} = 9 + 25 \\
\]
Next, we will further reduce the right-hand side terms as follows:
We can write 9 as \[{3^2}\] and 25 as \[{5^2}\].
Thus, replace the values obtained,
We get,
\[ \Rightarrow {3^{x - 1}} + {5^{x - 1}} = {3^2} + {5^2}\]
Further, on comparing the powers of the terms 3 and 5 on both sides of the expression, we get,
\[x - 1 = 2\] and \[x - 1 = 2\]
Thus, we will simplify the expression obtained above to evaluate the value of \[x\].
By keeping variables on one side and constants on the other side.
\[ \Rightarrow x = 1 + 2 = 3\]
Which shows that this question has only one solution.
Hence. Option (A) is correct.

Note: In this type of questions try to write the right-hand side in the form of powers having the same base as the left hand side, because if powers are not the same then we cannot compare the terms on both sides.