Answer :
To solve the exponential equation \(\left(\frac{1}{2}\right)^x = 6\), we need to find the value of \(x\) that makes the equation true. Here's a step-by-step process:
1. Rewrite the equation in logarithmic form:
[tex]\[\left(\frac{1}{2}\right)^x = 6\][/tex]
This can be transformed using logarithms. Recall that \(a^x = b\) is equivalent to \(x = \log_a(b)\). Therefore:
[tex]\[x = \log_{1/2}(6)\][/tex]
2. Convert the base of the logarithm:
Using the change of base formula for logarithms, \(\log_a(b) = \frac{\log_c(b)}{\log_c(a)}\) where \(c\) is any positive value (commonly 10 or \(e\)), we can write:
[tex]\[x = \frac{\log(6)}{\log(1/2)}\][/tex]
3. Evaluate the logarithms:
Plugging in the values (using natural logarithm in this case):
[tex]\[\log(6) \approx 1.792\][/tex]
[tex]\[\log(1/2) \approx -0.693\][/tex]
4. Calculate the value of \(x\):
[tex]\[x = \frac{1.792}{-0.693} \approx -2.585\][/tex]
5. Round to the nearest thousandth:
The value of \(x\) is already expressed to the nearest thousandth, so:
[tex]\[x \approx -2.585\][/tex]
Thus, the solution to the equation \(\left(\frac{1}{2}\right)^x = 6\) rounded to the nearest thousandth is:
[tex]\[x \approx -2.585\][/tex]
### Final Answer:
A. The solution set is [tex]\(\{-2.585\}\)[/tex].
1. Rewrite the equation in logarithmic form:
[tex]\[\left(\frac{1}{2}\right)^x = 6\][/tex]
This can be transformed using logarithms. Recall that \(a^x = b\) is equivalent to \(x = \log_a(b)\). Therefore:
[tex]\[x = \log_{1/2}(6)\][/tex]
2. Convert the base of the logarithm:
Using the change of base formula for logarithms, \(\log_a(b) = \frac{\log_c(b)}{\log_c(a)}\) where \(c\) is any positive value (commonly 10 or \(e\)), we can write:
[tex]\[x = \frac{\log(6)}{\log(1/2)}\][/tex]
3. Evaluate the logarithms:
Plugging in the values (using natural logarithm in this case):
[tex]\[\log(6) \approx 1.792\][/tex]
[tex]\[\log(1/2) \approx -0.693\][/tex]
4. Calculate the value of \(x\):
[tex]\[x = \frac{1.792}{-0.693} \approx -2.585\][/tex]
5. Round to the nearest thousandth:
The value of \(x\) is already expressed to the nearest thousandth, so:
[tex]\[x \approx -2.585\][/tex]
Thus, the solution to the equation \(\left(\frac{1}{2}\right)^x = 6\) rounded to the nearest thousandth is:
[tex]\[x \approx -2.585\][/tex]
### Final Answer:
A. The solution set is [tex]\(\{-2.585\}\)[/tex].
