What is the equation of the vertical asymptote of the func \( f(x)=\log _{9}(x+6) \) ?
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To find the vertical asymptote of the function \( f(x) = \log_{9}(x+6) \), we analyze when the argument of the logarithm becomes zero, since logarithms are undefined for non-positive values. Setting the argument \( x + 6 = 0 \), we find \( x = -6 \). Thus, the vertical asymptote occurs at \( x = -6 \), which means as \( x \) approaches -6 from the right, \( f(x) \) will decrease without bound. If you're trying to remember how to identify vertical asymptotes in logarithmic functions, just look for where the input of the log function equals zero! This is usually a straightforward method: find where the argument goes negative or zero, and voila, you’ve got your asymptote. So in this case, when \( x + 6 \) hits zero, you know you found it!