Question

Solve for x:

log (7x + 3) – log (x – 3) = 1

Answer 1

log (7x + 3) – log (x – 3) = 1

Using the rule log ⁡a − log ⁡b = `log ⁡(a/b)`,

`log((7x + 3)/(x - 3))` = 1

We know that 1 = log ⁡10  ....(Since the base is 10.)

`log ((7x + 3)/(x - 3))` = log 10

Since the logarithms are equal, their arguments are equal:

`(7x + 3)/(x - 2)` = 10

Now, multiply both sides by x − 3:

7x + 3 = 10(x – 3)

7x + 3 = 10x – 30

3 + 30 = 10x – 7x

33 = 3x

x = `33/3`

x = 11

Validity Check:

7x + 3 = 77 + 3 = 80 > 0

x – 3 = 11 – 3 = 8 > 0

All expressions valid.

x = 11