How do you solve log_(121)44 = x?

Dec 7, 2015

I found: $x = 0.789$

Explanation:

We can write (using the definition of log):
$44 = {121}^{x}$
then
$11 \cdot 4 = {11}^{2 x}$
taking $11$ to the right:
$4 = {11}^{2 x} / 11$
using the property of the quotient of exponents with the same base:
$4 = {11}^{2 x - 1}$
now we can take the natural log of both sides:
$\ln \left(4\right) = \ln \left({11}^{2 x - 1}\right)$
we can now use the property of the logs:
$\log {x}^{a} = a \log x$
to get:
$\ln 4 = \left(2 x - 1\right) \cdot \ln 11$
so that:
$2 x - 1 = \ln \frac{4}{\ln} \left(11\right)$
rearranging:
$x = \frac{1}{2} \left[\ln \frac{4}{\ln} \left(11\right) + 1\right] = 0.789$