To evaluate the expression [tex]\(\log_{19} 13\)[/tex] using either common logarithms (base 10) or natural logarithms (base [tex]\(e\)[/tex]), we can utilize the change-of-base formula.
The change-of-base formula states that for any positive numbers [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] (where [tex]\(a \neq 1\)[/tex] and [tex]\(b \neq 1\)[/tex]):
[tex]$
\log_a b = \frac{\log_c b}{\log_c a}
$[/tex]
In this case, we want to rewrite [tex]\(\log_{19} 13\)[/tex] using either common logarithms (base 10) or natural logarithms (base [tex]\(e\)[/tex]).
Let's rewrite it using common logarithms (base 10):
[tex]$
\log_{19} 13 = \frac{\log_{10} 13}{\log_{10} 19}
$[/tex]
Alternatively, we can rewrite it using natural logarithms (base [tex]\(e\)[/tex]):
[tex]$
\log_{19} 13 = \frac{\ln 13}{\ln 19}
$[/tex]
So, the expression [tex]\(\log_{19} 13\)[/tex] can be rewritten using the change-of-base property as:
[tex]$
\log_{19} 13 = \frac{\log 13}{\log 19}
$[/tex]
or
[tex]$
\log_{19} 13 = \frac{\ln 13}{\ln 19}
$[/tex]
Choose either form based on your preference for common logarithms or natural logarithms. Both forms are correct and equivalent.
