lnx-4ln3= ln(5/x)
Recall that alnb=ln(a^b), so
lnx-ln3^4=ln(5/x)
Recall that lna-lnb=ln(a/b), so
ln(x/3^4)= ln(5/x)
Convert the logarithmic equation to an exponential equation.
e^ln(x/3^4)= e^ln(5/x)
Remember that e^lnx =x, so
x/3^4=5/x
x^2=5×3^4
x=±sqrt(5×3^4) =±sqrt5×sqrt(3^4)=±sqrt5×3^2
x=9sqrt5 and x=-9sqrt5 are possible solutions.
Check:
lnx-4ln3= ln(5/x)
If x=9sqrt5,
ln(9sqrt5)-4ln3= ln(5/(9sqrt5))
ln9+lnsqrt5-ln3^4 =ln5-ln(9sqrt5)
ln9+lnsqrt5-ln(3^2)^2 =ln(sqrt5)^2-ln9-lnsqrt5
ln9+lnsqrt5-2ln9=2lnsqrt5-ln9-lnsqrt5
lnsqrt5-ln9=lnsqrt5-ln9
∴ x=9sqrt5 is a solution.
If x=-9sqrt5,
ln(-9sqrt5)-4ln3= ln(5/(-9sqrt5))=ln5-ln(-9sqrt5)
But ln(-9sqrt5) is not defined.
∴ x=-9sqrt5 is not a solution.