The equation given is:

\[ y = \log\left(\frac{1}{3}\right)^x \]

We can simplify this expression using logarithmic properties. One key property is that $\log(a^b) = b \cdot \log(a)$. Applying this, we get:

\[
y = x \cdot \log\left(\frac{1}{3}\right)
\]

Since $\log\left(\frac{1}{3}\right)$ is a constant, let's denote it by $c$:

\[
y = c \cdot x
\]

Where $c = \log\left(\frac{1}{3}\right)$.

The value of $\log\left(\frac{1}{3}\right)$ is negative because $\frac{1}{3}$ is a number between 0 and 1. Therefore, the equation represents a linear function with a negative slope.

Would you like to calculate the exact value of $c$ or explore further details?

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Here are 8 related questions you might find interesting:

1. What is the base of the logarithm used here?
2. How does changing the base of the logarithm affect the value of $c$?
3. What happens to the graph of $y = c \cdot x$ when $c$ is negative?
4. Can you plot the graph of $y = \log\left(\frac{1}{3}\right)^x$ for different values of $x$?
5. How does the value of $x$ affect $y$ when $c$ is negative?
6. What is the domain and range of the function $y = \log\left(\frac{1}{3}\right)^x$?
7. How would the equation change if the expression inside the logarithm were $\frac{3}{1}$ instead of $\frac{1}{3}$?
8. Can you explain the significance of the logarithmic function in real-world applications?

**Tip:** When working with logarithmic expressions, remember that $\log(a^b) = b \cdot \log(a)$ can simplify complex equations significantly.

Learn how to understand and solve the equation $ y = \log\left(\frac{1}{3}\right)^x $. Explore the properties of logarithms, exponential functions, and linear functions involved in this equation. Suitable for high school students and those interested in advanced algebraic concepts.

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