Question

Solve for x:

log (3x + 5) – log (x – 3) = 1

Answer 1

log (3x + 5) – log (x – 3) = 1

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

`log((3x + 5)/(x - 3))` = log 1

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

`log ((3x + 5)/(x - 3))` = log 10

Since the logarithms are equal, their arguments are equal:

`(3x + 5)/(x - 3)` = 10

Now, multiply both sides by x − 3:

3x + 5 = 10(x – 3)

3x + 5 = 10x – 30

5 + 30 = 10x – 3x

35 = 7x

x = `35/7`

x = 5

Validity Check:

⇒ 3x + 5

= 3 × 5 + 5

= 15 + 5

= 20 > 0

⇒ x – 3

= 5 – 3

= 2 > 0

All expressions are valid.

x = 5