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In graphing logarithmic functions, one of the key features to identify is the vertical asymptote. This represents a line that the graph approaches but never touches or crosses. For the function \( f(x) = \log(x+10) \), the vertical asymptote occurs where the argument of the logarithm is zero. To find this, set \( x+10 = 0 \), which results in \( x = -10 \). This means that as \( x \) approaches -10 from the right, the value of \( f(x) \) decreases toward negative infinity. This behavior occurs because a logarithm is undefined for negative values and zero, creating a boundary at the asymptote.
It is important to note that this vertical asymptote draws a line where the function has a significant change. As \( x \) gets closer to \( -10 \), the graph will dive sharply downwards, showcasing how the logarithmic function behaves near its vertical asymptote.

A logarithmic function is characterized by its slow growth as its base grows larger. In the function \( f(x) = \log(x+10) \), we are working with a common log where the base is \( 10 \). Logarithmic functions like this one increase quickly initially for small values of \( x \) after their shift rightward, then, their growth becomes gradual.
The domain of this function is \( x > -10 \), because the argument \( (x+10) \) must be a positive number. When x is less than -10, the function is undefined, reinforcing the location of the vertical asymptote. Understanding logarithmic functions is about recognizing their domains, ranges, and growth tendencies. They often appear in real-world scenarios that involve exponential growth or sound intensity, which highlights the importance of mastering their graphical representations.

The horizontal shift in a logarithmic function is controlled by the constant added to or subtracted from the \( x \) inside the logarithm. In our function \( f(x) = \log(x+10) \), we have a shift leftward by 10 units. This shift modifies the basic \( \log x \) function to move its graph left on the x-axis.
Understanding horizontal shifts is crucial in graphing, as the alteration affects not only where the graph starts, but also how the function's other attributes align on the graph. By shifting the base \( x \) in this manner, both the vertical asymptote and all related intercepts are moved accordingly, making the visualization of the function intuitive once you gain this perspective.

Intercepts provide key points where the graph crosses the axes. For \( f(x) = \log(x+10) \), let's first determine the x-intercept by setting \( f(x) = 0 \). Solving \( \log(x+10) = 0 \) gives us \( x+10 = 1 \), so \( x = -9 \). This point, \( (-9, 0) \), is where the function crosses the x-axis, which is critical for graphing accuracy.
It's worth noting that this function has no y-intercept because at \( x = 0 \), the logarithmic function outputs a number that is calculated from negative arguments within the log, landing outside of its domain. Thus, understanding the locations and existence of intercepts help to sketch the function with more precision, ensuring a clear and accurate graph depiction.