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Chapter 3: Problem 42

Find the domain, \(x\) -intercept, and vertical asymptote of the logarithmic function and sketch its graph. $$y=\log _{5}(x-1)+4$$

Short Answer

Expert verified
The domain of the function \( y=\log_{5}(x-1)+4 \) is \( (1, \infty) \). The x-intercept is at \( x = 1.00016 \). The vertical asymptote is at \( x = 1 \). The graph can be sketched based on these properties, rising from the left of the asymptote \( x = 1 \) and then continuing to rise as \( x \) increases, and it will be shifted four units upwards.

Step by step solution

01

Identify the Basic Function

The basic function here is \( y= \log_{5}(x) \), which means the base of the logarithm is 5 and \( y \) is written in terms of \( x \). It undergoes a transformation to become \( y=\log_{5}(x-1)+4 \), where 1 is subtracted from \( x \) in the logarithm and 4 is added to the logarithm.
02

Determine the Domain

For a function to have a real number as a logarithm, the argument of the logarithm must be greater than 0. So the domain of the function is every \( x \) such that \( x > 1 \). The domain in interval notation is \( (1, \infty) \).
03

Calculate the X-Intercept

The x-intercept is where the function crosses the x-axis, which is when \( y = 0 \). Setting up the equation: \( 0=\log_{5}(x-1)+4 \), we then subtract 4 from both sides to give: \( -4=\log_{5}(x-1) \). We then convert this logarithmic equation to an exponential equation to solve for \( x \). Doing this gives us \( x = 5^{-4} + 1 \approx 1.00016 \).
04

Identify the Vertical Asymptote

Vertical asymptote occurs at the value which is not in the domain of the function and is the boundary of the domain. As the domain of the function is \( x > 1 \), the vertical asymptote is at \( x = 1 \).
05

Sketch the Graph

Start by plotting the x-intercept at \( (1.00016, 0) \) and the vertical asymptote as a dashed line at \( x = 1 \). Since the base of the logarithm is greater than 1 and the function is increasing for \( x > 1 \), the graph will rise from the left of the asymptote and continue to rise as \( x \) increases. A shift up by four will be observed due to the +4 in the function.

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