- Determine the domain by solving the inequality \(5 - 9x > 0\), resulting in \(x < \frac{5}{9}\).
- Identify the vertical asymptote by setting the argument equal to zero: \(5 - 9x = 0\), which gives \(x = \frac{5}{9}\).
- Calculate the x-intercept by solving \(\log_3(5 - 9x) = 0\), resulting in \(x = \frac{4}{9}\).
- Final answers: Domain \(\boxed{(-\infty, \frac{5}{9})}\), Asymptote \(\boxed{x = \frac{5}{9}}\), x-intercept \(\boxed{x = \frac{4}{9}}\).
Explanation
- Problem Analysis
The function provided is \(f(x) = \log_3(5 - 9x)\). To analyze this function, we must consider the properties of logarithmic functions. A logarithmic function of the form \(y = \log_b(u)\) is defined only when the argument \(u\) is strictly greater than zero, i.e., \(u > 0\). Additionally, as the argument \(u\) approaches 0 from the positive side, the function approaches negative infinity, indicating the presence of a vertical asymptote at \(u = 0\). Finally, the x-intercept is the point where the output of the function is zero, which corresponds to the case where the argument \(u\) equals 1, since \(\log_b(1) = 0\).
- Determining the Domain
To find the domain, we require the argument \(5 - 9x\) to be positive:
\[5 - 9x > 0\]
Subtracting 5 from both sides gives \(-9x > -5\). Dividing both sides by -9 and remembering to reverse the inequality sign, we get:
\[x < \frac{5}{9}\]
Thus, the domain of the function is the interval \((-\infty, \frac{5}{9})\).
- Finding the Vertical Asymptote
The vertical asymptote occurs where the argument of the logarithm becomes zero. We solve:
\[5 - 9x = 0\]
Adding \(9x\) to both sides, we have \(5 = 9x\), which simplifies to:
\[x = \frac{5}{9}\]
The vertical asymptote is the vertical line \(x = \frac{5}{9}\).
- Finding the x-intercept
The x-intercept occurs when \(f(x) = 0\). This leads to:
\[\log_3(5 - 9x) = 0\]
By the definition of the logarithm, this implies:
\[5 - 9x = 3^0\]
Since \(3^0 = 1\), we have:
\[5 - 9x = 1\]
Subtracting 5 from both sides:
\[-9x = -4\]
Dividing by -9:
\[x = \frac{4}{9}\]
So, the x-intercept is at \(x = \frac{4}{9}\).
Examples
Logarithmic functions are used extensively in various fields to represent phenomena that vary over huge ranges of magnitude. For example, in chemistry, the pH scale, which measures acidity, is defined using a logarithm (\(pH = -\log[H^+]\)). Understanding how to find the domain and key features of such functions helps scientists interpret these scales accurately. Similarly, in fields like seismology or acoustics, logarithmic scales (like the Richter scale or decibels) are used to measure earthquake intensity or sound levels, allowing for the effective management of vast amounts of data.