Question

If \(11^{10-2x} = 1\), find the value of \(x\).

• A. 10
• B. 5
• C. 2
• D. None of these

Hint:

A nonzero base raised to the power 0 always gives 1, so set the exponent \(10-2x\) equal to 0.

Answer 1

The Correct Option is B

Step 1: Take log on both sides.
The equation is $11^{10-2x} = 1$. Instead of reasoning about the base, we apply a logarithm to both sides to bring the exponent down.

Step 2: Apply the log rule.
Taking log to base 11 on both sides gives $\log_{11}\left(11^{10-2x}\right) = \log_{11}(1)$.
Using the rule $\log_a(a^n) = n$, the left side simplifies to $10-2x$.
Using the rule $\log_a(1) = 0$ for any valid base $a$, the right side becomes $0$.

Step 3: Solve the resulting linear equation.
The equation reduces to $10 - 2x = 0$.
Rearranging, $2x = 10$, so $x = 5$.

Step 4: Verify by substitution.
Put $x = 5$ back into the exponent: $10 - 2(5) = 10 - 10 = 0$.
The expression becomes $11^{0}$, which equals 1, matching the right side of the original equation and confirming the answer.

Final Answer:
The value of x is confirmed to be 5. \[ \boxed{x = 5} \]