Math Problem Statement

y=\log\frac13 $$$$ x

Solution

The equation given is:

y=log(13)xy = \log\left(\frac{1}{3}\right)^x

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

y=xlog(13)y = x \cdot \log\left(\frac{1}{3}\right)

Since log(13)\log\left(\frac{1}{3}\right) is a constant, let's denote it by cc:

y=cxy = c \cdot x

Where c=log(13)c = \log\left(\frac{1}{3}\right).

The value of log(13)\log\left(\frac{1}{3}\right) is negative because 13\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 cc or explore further details?


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 cc?
  3. What happens to the graph of y=cxy = c \cdot x when cc is negative?
  4. Can you plot the graph of y=log(13)xy = \log\left(\frac{1}{3}\right)^x for different values of xx?
  5. How does the value of xx affect yy when cc is negative?
  6. What is the domain and range of the function y=log(13)xy = \log\left(\frac{1}{3}\right)^x?
  7. How would the equation change if the expression inside the logarithm were 31\frac{3}{1} instead of 13\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(ab)=blog(a)\log(a^b) = b \cdot \log(a) can simplify complex equations significantly.

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Math Problem Analysis

Mathematical Concepts

Logarithms
Exponential Functions
Linear Functions

Formulas

Logarithm properties: $\log(a^b) = b \cdot \log(a)$

Theorems

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Suitable Grade Level

High School