Single Variable Calculus, Chapter 7, Review Exercises, Section Review Exercises, Problem 10

If $a > 1$ for large values of $x$, which of the functions $y = x^a, y = a^x$ and $y = \log_a x$ has the largest values and which has smallest values?


$
\begin{equation}
\begin{aligned}

\text{Let } a =& 10, \text{ then}
\\
\\
\text{if } x =& 100, \text{ then}
\\
\\
y =& x^a = 100^{10} = 1x 10^{20},
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y =& a^x = 10^{100} = 1x 10^{100}, \text{ and}
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\\
y =& \log_a x = \log_{10} 100 = 2
\\
\\
\text{But if } x =& 10, \text{ then}
\\
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y =& x^a = 10^{10} = 1x 10^{10}
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y =& a^x = 10^{10} = 1x 10^{10}, \text{ and}
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\\
y =& \log_a x = \log_{10} 10 = 1


\end{aligned}
\end{equation}
$


Based from the values we've given, $y = a^x$ has the largest value and $y = \log_a x$ has the smallest value. But if the values of $a$ and $x$ are equal then $y = a^x$ and $y = x^a$ have the largest values and still $y = \log_a x$ has the smallest value.