To solve the problem [tex]\( 3^x = 11 \)[/tex] and [tex]\( 11^y = 81 \)[/tex] and find the value of [tex]\( x \times y \)[/tex], follow these steps:
1. Solve for [tex]\( x \)[/tex]:
- We have the equation [tex]\( 3^x = 11 \)[/tex].
- To solve for [tex]\( x \)[/tex], take the logarithm of both sides. Using the natural logarithm ([tex]\(\ln\)[/tex]), we have:
[tex]\[
\ln(3^x) = \ln(11)
\][/tex]
- Using the property of logarithms that [tex]\(\ln(a^b) = b \ln(a)\)[/tex], this becomes:
[tex]\[
x \ln(3) = \ln(11)
\][/tex]
- Isolate [tex]\( x \)[/tex] by dividing both sides by [tex]\(\ln(3)\)[/tex]:
[tex]\[
x = \frac{\ln(11)}{\ln(3)}
\][/tex]
2. Solve for [tex]\( y \)[/tex]:
- We have the equation [tex]\( 11^y = 81 \)[/tex].
- Similarly, take the logarithm of both sides:
[tex]\[
\ln(11^y) = \ln(81)
\][/tex]
- Using the property of logarithms, this becomes:
[tex]\[
y \ln(11) = \ln(81)
\][/tex]
- Isolate [tex]\( y \)[/tex] by dividing both sides by [tex]\(\ln(11)\)[/tex]:
[tex]\[
y = \frac{\ln(81)}{\ln(11)}
\][/tex]
3. Calculate [tex]\( x \times y \)[/tex]:
- Now that we have expressions for both [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
x = \frac{\ln(11)}{\ln(3)}
\][/tex]
[tex]\[
y = \frac{\ln(81)}{\ln(11)}
\][/tex]
- Multiply [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
x \times y = \left( \frac{\ln(11)}{\ln(3)} \right) \times \left( \frac{\ln(81)}{\ln(11)} \right)
\][/tex]
4. Simplify the expression:
- Notice that the [tex]\(\ln(11)\)[/tex] terms in the numerator and denominator cancel each other:
[tex]\[
x \times y = \frac{\ln(11) \times \ln(81)}{\ln(3) \times \ln(11)} = \frac{\ln(81)}{\ln(3)}
\][/tex]
- We know:
[tex]\[
81 = 3^4
\][/tex]
- Therefore:
[tex]\[
\ln(81) = \ln(3^4) = 4 \ln(3)
\][/tex]
- Substitute this back:
[tex]\[
x \times y = \frac{4 \ln(3)}{\ln(3)} = 4
\][/tex]
So, the product [tex]\( x \times y = 4 \)[/tex].
Thus, the correct answer is:
[tex]\[ \boxed{4} \][/tex]
