Problem 83 - Precalculus

Question

Graph the logarithmic function using transformation techniques. State the domain and range of \(f\). $$f(x)=\log (2 x)$$

Step-by-Step Solution

Answer

Domain: \((0, \infty)\); Range: \((-\infty, \infty)\)

Step 1: Identify the Parent Function

The parent function for the given logarithmic function is the logarithm base 10, represented as \( g(x) = \log(x) \). The graph of \( \log(x) \) passes through point \((1, 0)\) and has a vertical asymptote on the y-axis at \( x=0 \).

Step 2: Determine the Transformation

The given function \( f(x) = \log(2x) \) is a transformation of the parent function \( \log(x) \). The transformation involves a horizontal compression. The multiplication by 2 inside the logarithm compresses the graph horizontally by a factor of \frac{1}{2}, meaning any input \( x \) in the original graph is reduced to \( \frac{x}{2} \).

Step 3: Graph the Transformed Function

To graph \( f(x) = \log(2x) \), start with the basic positions of \( g(x) = \log(x) \). The vertical asymptote, initially at \( x=0 \), remains in place because the transformation is horizontal. Key points like \( (1, 0) \) on \( g(x) \) become \( (\frac{1}{2}, 0) \) on \( f(x) \). For a few more points, consider when \( 2x = 10, \) which results in a y-value of 1 at \( x = 5. \)

Step 4: Determine the Domain and Range

The domain of \( f(x) = \log(2x) \) is determined by where the argument of the logarithm is positive: \( 2x > 0 \) which simplifies to \( x > 0 \). Therefore, the domain of \( f(x) \) is \( (0, \infty) \). The range of any log function, including \( f(x) \), is all real numbers: \( (-\infty, \infty) \).

Key Concepts

Understanding the Domain and Range of Logarithmic Functions

In the context of logarithmic functions, the domain refers to all the permissible x-values that the function can take, while the range is the set of y-values that the function can produce.
For the function \( f(x) = \log(2x) \), the domain is determined by ensuring the expression inside the logarithm remains positive.
Here, \( 2x > 0 \) simplifies to \( x > 0 \), indicating that the function is defined for all x greater than 0.
Thus, the domain is \( (0, \infty) \).

The range of logarithmic functions like \( f(x) \) is usually the entire set of real numbers.
This is because as x becomes larger, the log function increases without bound, and as x approaches the vertical asymptote from the right, the function decreases infinitely.

• Domain: \( (0, \infty) \)
• Range: \( (-\infty, \infty) \)

Exploring Graph Transformations in Logarithmic Functions

Graph transformations are tricks to modify a parent function into a more complex shape by applying specific operations.
With logarithmic functions such as \( f(x) = \log(2x) \), these transformations help us understand the function's behavior.

The transformation from \( g(x) = \log(x) \) to \( f(x) = \log(2x) \) involves a horizontal change.
This transformation keeps the general shape of the logarithmic curve but modifies the graph's horizontal spread.
The vertical asymptote remains at \( x=0 \).

• Reflection: None in this function.
• Translation: No shift up, down, left, or right.
• Horizontal Compression: Caused by the multiplication by 2, squeezing the graph horizontally.

The Effect of Horizontal Compression on Logarithmic Functions

Horizontal compression is a specific type of graph transformation affecting how wide or narrow a graph appears.
In the function \( f(x) = \log(2x) \), horizontal compression is influenced by the factor of 2 inside the log.

Normally, if you have \( g(x) = \log(x) \), the logarithm function spreads out in its arc-like shape.
However, multiplying x by 2 compresses the graph horizontally by a factor of \( \frac{1}{2} \).
This compression means each x-value is effectively halved.

For example, if \( g(x) \) passes through \((1, 0)\), \( f(x) \) passes through \(\left( \frac{1}{2}, 0 \right) \).
If you previously had a point at \( x = 10 \), the transformed point is at \( x = 5 \).

Horizontal compression maintains the same range of y-values while altering the domain's appearance.
This adjustment makes understanding transformation on logarithmic functions essential.

• Compression Factor: \( \frac{1}{2} \)
• Effect on x-values: Each x-value is halved.
• Graph Shape: Maintains the standard log curve but in a "squeezed" form.