find the set of all solutions of the equation ln(4x-1)=5-ln(x)?

Answer 1

ln(4x - 1) = 5 - ln(x)

First, move all logs to the left hand side.

ln(4x - 1) + ln(x) = 5

Use the log identities to combine the two logs; in particular,
log[base b](a) + log[base b](c) = log[base b](ac)

ln ( (4x - 1)*x ) = 5

Convert to exponential form, to obtain

e^5 = (4x - 1)x

Solve for x.

e^5 = 4x^2 - x

Move everything to the right hand side,

0 = 4x^2 - x - e^5 = 0

Solve using the quadratic formula.

x = [ 1 +/- sqrt( (-1)^2 - 4(4)(-e^5) ] / (2*4)

x = [ 1 +/- sqrt(1 + 16e^5) ] / 8

Answer 2

ln (4x - 1) + ln x = 5
ln [(4x - 1) (x) ] = 5
(4x - 1) (x) = e^5
4x² - x - e^5 = 0
4x² - x - 148.3 = 0
x = [ 1 ± √(1 + 2373) ] / 8
x = [1 ± 48.7] / 8
x = 6.2 , x = - 5.96

Answer 3

+ 1 to each side
then
4x=6
then
divide each side by six
1x=1.5
x=1.5