Answer :
To solve the equation \(2^x = 6\) for \(x\), we need to find the value of \(x\) that satisfies this equation.
First, we recognize that this equation doesn't have a solution that can be expressed as a simple rational number. Hence, we seek an approximate irrational solution. We can use logarithms to solve for \(x\).
Here are the steps:
1. Take the natural logarithm (or logarithm to any base, but commonly the natural logarithm is used) on both sides of the equation:
[tex]\[ \ln(2^x) = \ln(6) \][/tex]
2. Using the logarithmic property \(\ln(a^b) = b \ln(a)\), we can rewrite the left side:
[tex]\[ x \cdot \ln(2) = \ln(6) \][/tex]
3. Solve for \(x\) by dividing both sides by \(\ln(2)\):
[tex]\[ x = \frac{\ln(6)}{\ln(2)} \][/tex]
4. Now, calculate the value numerically. Using a calculator or logarithmic values:
[tex]\[ x \approx \frac{\ln(6)}{\ln(2)} \approx 2.5849609375 \][/tex]
Therefore, the approximate solution to the equation \(2^x = 6\) is:
[tex]\[ x \approx 2.5850 \][/tex]
So, the correct choice is:
A. The solution set is \{ [tex]\(2.5850\)[/tex] \}. (Rounded to four decimal places.)
First, we recognize that this equation doesn't have a solution that can be expressed as a simple rational number. Hence, we seek an approximate irrational solution. We can use logarithms to solve for \(x\).
Here are the steps:
1. Take the natural logarithm (or logarithm to any base, but commonly the natural logarithm is used) on both sides of the equation:
[tex]\[ \ln(2^x) = \ln(6) \][/tex]
2. Using the logarithmic property \(\ln(a^b) = b \ln(a)\), we can rewrite the left side:
[tex]\[ x \cdot \ln(2) = \ln(6) \][/tex]
3. Solve for \(x\) by dividing both sides by \(\ln(2)\):
[tex]\[ x = \frac{\ln(6)}{\ln(2)} \][/tex]
4. Now, calculate the value numerically. Using a calculator or logarithmic values:
[tex]\[ x \approx \frac{\ln(6)}{\ln(2)} \approx 2.5849609375 \][/tex]
Therefore, the approximate solution to the equation \(2^x = 6\) is:
[tex]\[ x \approx 2.5850 \][/tex]
So, the correct choice is:
A. The solution set is \{ [tex]\(2.5850\)[/tex] \}. (Rounded to four decimal places.)
