# If log 5a - log (2a-3) = 1, find a.

### 2 Answers | Add Yours

Given the logarithm equation:

log 5a - log (2a-3) = 1

We need to find the value of "a" that satisfies the equation.

We will use the logarithm properties to solve.

We know that log a - log b = log (a/b)

==> log 5a - log (2a-3) = log (5a/(2a-3) = 1

Also, we know that log 10 = 1

==> log 5a/(2a-3) = log 10

Now that we have the logs are equal. then the bases are equal too.

==> 5a/(2a-3) = 10

We will multiply by 2a-3 both sides.

==> 5a = 10(2a-3)

==> 5a = 20a - 30

==> -15a = -30

We will divide by -15

**==> a = 2**

log 5a - log (2a-3) = 1, find a.

=> log{ 5a/(2a-3)} = 1 = log10, as log a-logb = log(a/b).

=> log{5a/(2a-3)} = log10.

We take antilog :

5a/(2a-3) = 10.

We multiply by (2a-3):

5a = 10(2a-3) = 20a-30

5a+30 = 20a.

30 = 20a-5a.

30 = 15a.

30/15 = 15a/a.

2= a.

Or a = 2.

Therefore a= 2.