To determine and describe the graph of the function [tex]\( y = 5 \log(x+3) \)[/tex], we can follow several steps to analyze and better understand its properties.
### Understanding the Function
1. Logarithm Basics:
- The logarithmic function [tex]\(\log(x)\)[/tex] is the inverse of the exponential function. For [tex]\( y = \log_b(x) \)[/tex], [tex]\( b^y = x \)[/tex].
- The natural logarithm, denoted [tex]\(\log(x) = \ln(x)\)[/tex], has the base [tex]\( e \approx 2.718 \)[/tex].
2. Function Transformation:
- [tex]\( y = 5 \log(x+3) \)[/tex] involves a horizontal shift of the basic logarithmic function [tex]\(\log(x)\)[/tex] to the left by 3 units.
- Furthermore, the [tex]\( 5 \)[/tex] in front of the log indicates a vertical stretching of the graph by a factor of 5.
### Key Characteristics
1. Domain:
- Since logarithms are only defined for positive arguments, [tex]\( x + 3 > 0 \)[/tex] which simplifies to [tex]\( x > -3 \)[/tex].
- Therefore, the domain of the function is [tex]\( x > -3 \)[/tex].
2. Vertical Asymptote:
- As [tex]\( x \)[/tex] approaches [tex]\( -3 \)[/tex] from the right, [tex]\( \log(x + 3) \)[/tex] tends to [tex]\(-\infty\)[/tex]. Thus, the function has a vertical asymptote at [tex]\( x = -3 \)[/tex].
3. Intercepts:
- Y-Intercept: When [tex]\( x = 0 \)[/tex],
[tex]\[
y = 5 \log(0 + 3) = 5 \log 3
\][/tex]
So the y-intercept is at [tex]\( \left( 0, 5 \log 3 \right) \)[/tex].
- X-Intercept: To find the x-intercept, set [tex]\( y = 0 \)[/tex].
[tex]\[
0 = 5 \log(x + 3) \implies \log(x + 3) = 0 \implies x + 3 = 1 \implies x = -2
\][/tex]
So the x-intercept is at [tex]\( x = -2 \)[/tex].
4. Behavior as [tex]\( x \to \infty \)[/tex]:
- As [tex]\( x \)[/tex] increases, [tex]\( \log(x + 3) \)[/tex] also increases but at a decreasing rate. Thus, [tex]\( y \to \infty \)[/tex].
### Graph Description
1. Shape and Orientation:
- The graph begins very close to the vertical asymptote at [tex]\( x = -3 \)[/tex], it sharply rises at first.
- It gradually curves upwards as [tex]\( x \)[/tex] increases due to the logarithmic nature, stretching vertically by a factor of 5.
- It crosses the x-axis at [tex]\( x = -2 \)[/tex] and continues to rise slowly.
2. Example Points:
- [tex]\( x = -2 \)[/tex] yields [tex]\( y = 0 \)[/tex]
- [tex]\( x = 0 \)[/tex] yields [tex]\( y = 5 \log 3 \approx 5 \times 1.0986 = 5.493 \)[/tex]
### Summary
The graph of [tex]\( y = 5 \log(x+3) \)[/tex]:
- Has a vertical asymptote at [tex]\( x = -3 \)[/tex].
- Has an x-intercept at [tex]\( x = -2 \)[/tex] and a y-intercept at [tex]\( (0, 5 \log 3) \)[/tex].
- Gradually increases towards infinity as [tex]\( x \)[/tex] grows larger.
- Reflects the properties of the logarithmic function, with a vertical stretch by a factor of 5.
Visualizing this, the graph starts near [tex]\( x = -3 \)[/tex], steeply rises at first, flattens out as [tex]\( x \)[/tex] increases, crosses the x-axis at [tex]\( x = -2 \)[/tex], and continues to rise slowly with increasing [tex]\( x \)[/tex].
