Verified answer

Never divide factors, you lose information

instead, remove the sqrt by squaring both sides

log(5x-1) = (log(5x-1))^2

log(5x-1) - (log(5x-1))^2 = 0

log(5x-1) (1-log(5x-1)) = 0

log(5x-1) = 0 or 1-log(5x-1) = 0

log(5x-1) = 0 or log(5x-1) = 1

log(1) = 0 and log(10) = 1 so

5x-1 = 1 or 5x-1 = 10

5x = 2 or 5x = 11

x = 2/5 or 11/5

There are TWO answers! When you divide out factors, you lose one of them.

u = log(5x-1)

u = u^2

0 = u(u-1)

u = 0 or 1

0 = log(5x-1)

1 = 5x-1

x = 2/5

1 = log(5x-1)

10 = 5x-1

x = 11/5

x = 2/5 and 11/5

"the radical is over the entire left expression only"

The clearest way to indicate this is the use parentheses around the radicand.

√(log(5x - 1)) = log(5x - 1)

log(5x - 1) = 1

5x - 1 = 10

5x = 11

x = 11/5

ie log (5x - 1) = 1

if base 10;

5x - 1 = 10

5x = 11

x = 11/5

if base e:

5x - 1 = e

x = (e + 1)/5