https://file.aiquickdraw.com/mathbot/file/17276247658494adzhv9w.jpg

The problem presented in the image is:

\[
\frac{9}{2} \cdot \log_5 x = -2
\]

We want to solve for \( x \). Here's a step-by-step solution:

### Step 1: Isolate the logarithmic expression
Divide both sides by \( \frac{9}{2} \):

\[
\log_5 x = \frac{-2}{\frac{9}{2}} = -\frac{2}{1} \cdot \frac{2}{9} = -\frac{4}{9}
\]

### Step 2: Convert the logarithmic equation to its exponential form
The equation \(\log_5 x = -\frac{4}{9}\) implies:

\[
x = 5^{-\frac{4}{9}}
\]

### Step 3: Exact solution
The exact value of \( x \) is:

\[
x = 5^{-\frac{4}{9}}
\]

### Step 4: Numerical approximation
We can find the approximate value using a calculator:

\[
x \approx 0.6207
\]

So, the solution is:

\[
x \approx 0.6207
\]

Would you like more details, or do you have any questions?

### Related questions:
1. How would the solution change if the base of the logarithm was 10 instead of 5?
2. What would happen if the equation was \( \frac{9}{2} \cdot \log_5 x = 2 \)?
3. How do we solve \( \log_2 (3x) = 4 \)?
4. How do you express \( \log_a b = c \) in its exponential form?
5. How do you simplify \( a^{\frac{m}{n}} \)?

### Tip:
When solving logarithmic equations, remember to convert between logarithmic and exponential forms for easier computation.

This page explains how to solve the logarithmic equation (9/2) * log_5(x) = -2. The step-by-step solution includes isolating the logarithm, converting to exponential form, and finding the approximate value of x.

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