We are given the equation:

$\log_8 x - \log_9 x = 1$

To solve this, we can use properties of logarithms. First, we apply the change of base formula to rewrite the logarithms with a common base. The change of base formula is:

$\log_b a = \frac{\log a}{\log b}$

Let's convert both logarithms to base 10 (or natural logarithms, but base 10 will work fine here):

$\log_8 x = \frac{\log x}{\log 8}, \quad \log_9 x = \frac{\log x}{\log 9}$

Now substitute these into the original equation:

$\frac{\log x}{\log 8} - \frac{\log x}{\log 9} = 1$

Next, factor out $\log x$ from the terms on the left-hand side:

$\log x \left( \frac{1}{\log 8} - \frac{1}{\log 9} \right) = 1$

Now, simplify the terms in parentheses. To do this, we need a common denominator:

$\frac{1}{\log 8} - \frac{1}{\log 9} = \frac{\log 9 - \log 8}{\log 8 \cdot \log 9}$

Using the property $\log a - \log b = \log \left( \frac{a}{b} \right)$, we get:

$\log 9 - \log 8 = \log \left( \frac{9}{8} \right)$

Substitute this back:

$\log x \left( \frac{\log \left( \frac{9}{8} \right)}{\log 8 \cdot \log 9} \right) = 1$

Now, solve for $\log x$:

$\log x = \frac{\log 8 \cdot \log 9}{\log \left( \frac{9}{8} \right)}$

Finally, raise 10 to the power of both sides to solve for $x$:

$x = 10^{\frac{\log 8 \cdot \log 9}{\log \left( \frac{9}{8} \right)}}$

Would you like me to compute a numerical approximation for $x$, or would you prefer a deeper explanation of this solution?

Here are a few related questions for deeper exploration:

1. How can we simplify logarithmic expressions with different bases?
2. What are the properties of logarithms that help in solving such equations?
3. How does the change of base formula help when working with logarithms of different bases?
4. What is the meaning of the logarithmic function in the context of exponential growth and decay?
5. How can we solve logarithmic equations involving both logarithmic and exponential terms?

Tip: When solving logarithmic equations, it's often helpful to first express all logarithms in terms of a common base to simplify calculations!