3^х=7^х/2
To solve the equation \( 3^x = \frac{7^x}{2} \), we can start by rearranging it:
Multiply both sides by 2 to eliminate the fraction: \[ 2 \cdot 3^x = 7^x \]
Next, let's express both sides with the same base if possible. However, since \( 3 \) and \( 7 \) don't share a common base, we can take the logarithm on both sides (we can use natural logarithm \( \ln \) or common logarithm \( \log \)). Here, I’ll use natural logarithm \( \ln \):
\[ \ln(2 \cdot 3^x) = \ln(7^x) \]
Using the properties of logarithms (specifically \( \ln(a \cdot b) = \ln a + \ln b \) and \( \ln(a^b) = b \cdot \ln a \)), we can rewrite the equation:
\[ \ln 2 + x \ln 3 = x \ln 7 \]
Now, to isolate \( x \), move all terms involving \( x \) to one side:
\[ \ln 2 = x \ln 7 - x \ln 3 \]
\[ \ln 2 = x (\ln 7 - \ln 3) \]
Now, divide both sides by \( \ln 7 - \ln 3 \) (assuming \( \ln 7 \neq \ln 3 \)):
\[ x = \frac{\ln 2}{\ln 7 - \ln 3} \]
This gives the value of \( x \) as:
\[ x \approx \frac{\ln 2}{\ln 7 - \ln 3} \]
You can calculate this value using a scientific calculator or computational tool:
- Using \( \ln 2 \approx 0.693 \)
- \( \ln 7 \approx 1.946 \)
- \( \ln 3 \approx 1.099 \)
So:
\[ x \approx \frac{0.693}{1.946 - 1.099} \approx \frac{0.693}{0.847} \approx 0.819 \]
Thus, the approximate solution for \( x \) is \( x \approx 0.819 \).
