Answer
$log 50^{-1}$
Work Step by Step
The smaller a base of a log, the greater the result.
$log 50^{-1}$
$log (1/50)$
$log .02$
$10^x=.02$
We know if $x=-1$, $10^x =.1$. If $x=-2$, however, $10^x=.01$. Thus, we know $-2 < x < -1$.
$ln 50^{-1}$
$ln .02$
$ln .02 =x$
$e^x=.02$
If $x=0$, $e^x=1$. If $x=-1$, $e^x=e^{-1} = 1/e$.
$1/e = 1/2.7 = 10/27 = .37$
If $x=-2$, $e^x = 1/(e^2)$
$1/(e^2) = 1/7.29$, which is between $.125$ ($1/8$) and $.14$ ($1/7$). This is not close to our value of $x$, so we know $x < -2$.
If $x=-3$, $e^x= 1/(e^3)$, and if $x=-4$, $e^x= 1/(e^-4)$.
$1/(e^3)$ is between $1/(2.5^3)$ and $1/(3^3)$. These values, respectively, are $1/15.625$ and $1/27$.
$1/15.625 = 2/31.25 = 4/62.5 = 8/125 = 64/1000 = .064$
$1/27$ is between $1/25 (.04)$ and $1/30 (.033)$
We are getting closer to our value of $.02$, so we know $log 50^{-1} > ln 50^{-1}$.
