Problem 5 - Precalculus Functions and Graphs

Question

Fill in the blank. The \(y\) -axis is a(n) _____________ for the graph of \(f(x)=\log _{a}(x)\).

Step-by-Step Solution

Answer

vertical asymptote

Step 1: Identify the equation type

The given equation is a logarithmic function in the form of \( f(x) = \log_a(x) \).

Step 2: Recall the properties of the logarithmic function

Recall that for the function \( f(x) = \log_a(x) \), the graph does not intersect the y-axis because the logarithm is undefined for non-positive values of x.

Step 3: Understand the behavior near the y-axis

As \( x \) approaches zero from the positive side, \( f(x) = \log_a(x) \) approaches negative infinity. This indicates that the y-axis acts as a boundary the graph approaches but never touches.

Step 4: Define the asymptote

Based on the behavior near the y-axis and the definition of an asymptote, the y-axis is a vertical asymptote of the logarithmic function \( f(x) = \log_a(x) \).

Key Concepts

Asymptote

When studying logarithmic functions like \(f(x) = \log_a(x)\), it's important to understand what an asymptote is. An asymptote is a line that the graph of a function approaches but never actually touches. In the case of \(f(x) = \log_a(x)\), the y-axis (\(x=0\)) is a vertical asymptote.
Why? As x gets very close to zero from the positive side, \(\log_a(x)\) heads towards negative infinity. This means that the graph will get very close to the y-axis as x decreases, but will never actually reach or intersect the axis. Therefore, the y-axis acts as a boundary that the logarithmic function approaches indefinitely.

Logarithm Properties

To understand logarithmic functions deeply, you'll need to get familiar with some fundamental logarithm properties. Here are a few key properties:

• Base Relationship: \( \log_a(a) = 1 \) because the log represents the exponent needed to achieve the base value.
• Product Rule: \( \log_a(xy) = \log_a(x) + \log_a(y) \) breaks down products into sums.
• Quotient Rule: \( \log_a \left( \frac{x}{y} \right) = \log_a(x) - \log_a(y) \) helps to simplify divisions into differences.
• Power Rule: \( \log_a(x^k) = k \cdot \log_a(x) \) which pulls power exponents out front.
• Change of Base Formula: \( \log_a(x) = \frac{\log_b(x)}{\log_b(a)} \) allows conversion between different log bases.

Understanding these properties will help in manipulating and solving equations involving logs.

Undefined Values

One significant property of logarithmic functions, such as \( f(x) = \log_a(x) \), is that they are only defined for positive values of x. This means for any \( x \leq 0 \), \( \log_a(x) \) is undefined.
Why does this happen? Remember, the logarithm \( \log_a(x) \) is the exponent to which the base \( a \) must be raised to produce \( x \). There is no real number solution for this when \( x \) is zero or negative. Since no power of a positive base number can result in zero or a negative number, the log of such values does not exist.
Thus, always ensure x is positive when dealing with logarithmic functions to avoid undefined expressions.