How do you solve ln x - 4 ln 3 = ln (5 / x)?

1 Answer
Aug 13, 2015

color(red)(x=9sqrt5)

Explanation:

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.