To find the value of \( u \) in the equation \( 7^u = \frac{3}{2} \), we can use logarithms. Here's the step-by-step solution:

1. Start with the given equation:
   \[
   7^u = \frac{3}{2}
   \]

2. Take the natural logarithm (ln) of both sides:
   \[
   \ln(7^u) = \ln\left(\frac{3}{2}\right)
   \]

3. Use the property of logarithms that allows you to bring the exponent in front:
   \[
   u \ln(7) = \ln\left(\frac{3}{2}\right)
   \]

4. Solve for \( u \) by dividing both sides by \( \ln(7) \):
   \[
   u = \frac{\ln\left(\frac{3}{2}\right)}{\ln(7)}
   \]

5. Calculate the values of the logarithms:
   \[
   \ln\left(\frac{3}{2}\right) \approx 0.4055
   \]
   \[
   \ln(7) \approx 1.9459
   \]

6. Plug these values into the equation:
   \[
   u \approx \frac{0.4055}{1.9459} \approx 0.2083
   \]

So, the value of \( u \) is approximately \( 0.2083 \).

Would you like more details or have any questions?

Here are 5 more questions you might want to ask:

1. How do you simplify logarithmic expressions?
2. What is the change of base formula for logarithms?
3. Can you solve exponential equations using other bases?
4. How do you solve logarithmic equations?
5. What are the properties of natural logarithms?

**Tip:** When dealing with exponential equations, logarithms are a powerful tool to isolate the variable and solve the equation.

Learn how to solve the exponential equation 7^u = 3/2 using logarithms with a detailed step-by-step solution. This problem involves exponential equations and logarithmic properties, suitable for high school students.

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