To solve the equation \((2^x) \times (5^x) = (3^x) \times 2\), we can start by simplifying both sides.
We can rewrite \((2^x) \times (5^x)\) as \((2 \times 5)^x = 10^x\).
So our equation now looks like: \[ 10^x = 3^x \times 2 \]
Next, we can divide both sides by \(3^x\): \[ \frac{10^x}{3^x} = 2 \]
This can be rewritten as: \[ \left(\frac{10}{3}\right)^x = 2 \]
Now we can take the logarithm of both sides. Let's use the natural logarithm (\(\ln\)): \[ \ln\left(\left(\frac{10}{3}\right)^x\right) = \ln(2) \]
Using the power rule for logarithms, this becomes: \[ x \cdot \ln\left(\frac{10}{3}\right) = \ln(2) \]
Now, solve for \(x\): \[ x = \frac{\ln(2)}{\ln\left(\frac{10}{3}\right)} \]
This is the solution for \(x\). You can approximate the value using a calculator if needed.
Calculating the values: \[ \ln(2) \approx 0.6931 \] \[ \ln\left(\frac{10}{3}\right) = \ln(10) - \ln(3) \approx 2.3026 - 1.0986 \approx 1.2040 \]
So, \[ x \approx \frac{0.6931}{1.2040} \approx 0.575 \]
Thus, the approximate solution for \(x\) is: \[ \boxed{0.575} \]