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To understand the graph of the function [tex]\( y = 5 \log(x + 3) \)[/tex], let's break down the components and characteristics step by step:
1. Understanding the Logarithm Function:
- The logarithmic function [tex]\( \log(x) \)[/tex] is only defined for [tex]\( x > 0 \)[/tex]. It has a vertical asymptote at [tex]\( x = 0 \)[/tex] and increases slowly without bound as [tex]\( x \)[/tex] increases.
- The natural logarithm [tex]\( \log(x) \)[/tex] refers to the logarithm with base [tex]\( e \)[/tex], where [tex]\( e \)[/tex] is approximately 2.718.
2. Transformation [tex]\( x + 3 \)[/tex]:
- Shifting the input of the log function by 3 units to the left, i.e., [tex]\( \log(x+3) \)[/tex], means the function inside the logarithm becomes positive when [tex]\( x > -3 \)[/tex].
- Therefore, the domain of the function [tex]\( y = 5 \log(x + 3) \)[/tex] is [tex]\( x > -3 \)[/tex].
3. Vertical Stretch by a Factor of 5:
- Multiplying the logarithmic function by 5 vertically stretches the graph by a factor of 5. This means every [tex]\( y \)[/tex]-value is expanded 5 times further away from the [tex]\( x \)[/tex]-axis.
4. Plotting Key Points:
- Let's determine a few values for better understanding:
- At [tex]\( x = -2 \)[/tex]:
[tex]\( y = 5 \log(-2 + 3) = 5 \log(1) = 5 \cdot 0 = 0 \)[/tex].
- At [tex]\( x = 0 \)[/tex]:
[tex]\( y = 5 \log(0 + 3) = 5 \log(3) \)[/tex].
Since [tex]\( \log(3) \approx 1.1 \)[/tex], [tex]\( y \approx 5 \times 1.1 = 5.5 \)[/tex].
- At [tex]\( x = 1 \)[/tex]:
[tex]\( y = 5 \log(1 + 3) = 5 \log(4) \)[/tex].
Since [tex]\( \log(4) \approx 1.386 \)[/tex], [tex]\( y \approx 5 \times 1.386 = 6.93 \)[/tex].
5. Asymptotic Behavior:
- The function [tex]\( y = 5 \log(x + 3) \)[/tex] has a vertical asymptote at [tex]\( x = -3 \)[/tex]. As [tex]\( x \)[/tex] approaches [tex]\( -3 \)[/tex] from the right, [tex]\( y \)[/tex] decreases without bound.
Using these points and characteristics, we can sketch the graph:
- The graph will start very high (approaching [tex]\(-\infty\)[/tex]) on the left slightly to the right of [tex]\( x = -3 \)[/tex] and will cross the [tex]\( y \)[/tex]-axis at [tex]\( (0, 5 \log(3)) \)[/tex], which is [tex]\( (0, 5.5) \)[/tex].
- It will then increase without bound as [tex]\( x \)[/tex] increases further to the right.
To illustrate the graph:
1. Draw the vertical asymptote at [tex]\( x = -3 \)[/tex].
2. Mark the points: [tex]\( (-2,0) \)[/tex], [tex]\( (0, 5.5) \)[/tex], and a few others.
3. Sketch the curve starting from the vertical asymptote [tex]\( x = -3 \)[/tex], passing through these points, and increasing slowly to the right.
In conclusion, the graph of [tex]\( y = 5 \log(x + 3) \)[/tex] is a logarithmic curve that has been shifted 3 units to the left and stretched vertically by a factor of 5. It has a vertical asymptote at [tex]\( x = -3 \)[/tex], passes through point [tex]\( (-2, 0) \)[/tex], and increases slowly as [tex]\( x \)[/tex] moves to the right.
1. Understanding the Logarithm Function:
- The logarithmic function [tex]\( \log(x) \)[/tex] is only defined for [tex]\( x > 0 \)[/tex]. It has a vertical asymptote at [tex]\( x = 0 \)[/tex] and increases slowly without bound as [tex]\( x \)[/tex] increases.
- The natural logarithm [tex]\( \log(x) \)[/tex] refers to the logarithm with base [tex]\( e \)[/tex], where [tex]\( e \)[/tex] is approximately 2.718.
2. Transformation [tex]\( x + 3 \)[/tex]:
- Shifting the input of the log function by 3 units to the left, i.e., [tex]\( \log(x+3) \)[/tex], means the function inside the logarithm becomes positive when [tex]\( x > -3 \)[/tex].
- Therefore, the domain of the function [tex]\( y = 5 \log(x + 3) \)[/tex] is [tex]\( x > -3 \)[/tex].
3. Vertical Stretch by a Factor of 5:
- Multiplying the logarithmic function by 5 vertically stretches the graph by a factor of 5. This means every [tex]\( y \)[/tex]-value is expanded 5 times further away from the [tex]\( x \)[/tex]-axis.
4. Plotting Key Points:
- Let's determine a few values for better understanding:
- At [tex]\( x = -2 \)[/tex]:
[tex]\( y = 5 \log(-2 + 3) = 5 \log(1) = 5 \cdot 0 = 0 \)[/tex].
- At [tex]\( x = 0 \)[/tex]:
[tex]\( y = 5 \log(0 + 3) = 5 \log(3) \)[/tex].
Since [tex]\( \log(3) \approx 1.1 \)[/tex], [tex]\( y \approx 5 \times 1.1 = 5.5 \)[/tex].
- At [tex]\( x = 1 \)[/tex]:
[tex]\( y = 5 \log(1 + 3) = 5 \log(4) \)[/tex].
Since [tex]\( \log(4) \approx 1.386 \)[/tex], [tex]\( y \approx 5 \times 1.386 = 6.93 \)[/tex].
5. Asymptotic Behavior:
- The function [tex]\( y = 5 \log(x + 3) \)[/tex] has a vertical asymptote at [tex]\( x = -3 \)[/tex]. As [tex]\( x \)[/tex] approaches [tex]\( -3 \)[/tex] from the right, [tex]\( y \)[/tex] decreases without bound.
Using these points and characteristics, we can sketch the graph:
- The graph will start very high (approaching [tex]\(-\infty\)[/tex]) on the left slightly to the right of [tex]\( x = -3 \)[/tex] and will cross the [tex]\( y \)[/tex]-axis at [tex]\( (0, 5 \log(3)) \)[/tex], which is [tex]\( (0, 5.5) \)[/tex].
- It will then increase without bound as [tex]\( x \)[/tex] increases further to the right.
To illustrate the graph:
1. Draw the vertical asymptote at [tex]\( x = -3 \)[/tex].
2. Mark the points: [tex]\( (-2,0) \)[/tex], [tex]\( (0, 5.5) \)[/tex], and a few others.
3. Sketch the curve starting from the vertical asymptote [tex]\( x = -3 \)[/tex], passing through these points, and increasing slowly to the right.
In conclusion, the graph of [tex]\( y = 5 \log(x + 3) \)[/tex] is a logarithmic curve that has been shifted 3 units to the left and stretched vertically by a factor of 5. It has a vertical asymptote at [tex]\( x = -3 \)[/tex], passes through point [tex]\( (-2, 0) \)[/tex], and increases slowly as [tex]\( x \)[/tex] moves to the right.
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