Question

How do you solve `Log x + log(x+3)= log 10`?

Answer 1

Use properties of `log`, exponentiation and solve a quadratic to find solution `x=2`.

Explanation:

`log(a)+log(b) = log(ab)`

So `log x + log(x+3) = log(x(x+3)) = log(x^2+3x)`

Note we require `x > 0` in order that `log x` is defined.

So:

`log(x^2+3x) = log 10`

Take exponent base `10` of both sides to get:

`x^2 + 3x = 10`

Subtract `10` from both sides to get:

`0 = x^2+3x-10 = (x+5)(x-2)`

So `x=-5` or `x=2`.

`x=-5` is spurious, because we require `x > 0` in order that `log x` be defined.

So the only valid solution is `x=2`