Checking whether answers of logarithmic and exponential equalities are correct.

Question

When you check the answers you get from equalities like for example:

$$ ^2\log(x-2) = 3- ^2\log(x)$$

$$ 4^x = 3 \times 2^x + 10$$

so on and so forth, is it sufficient to do the following:

• For the $x$'es you get from the logarithmic one, just plug them in $(x-2)$ and $(x)$ and see if it is $>0$.

• For the exponential one, check if $x > 0$.

Or do you actually need to plug in your answers in the equality and check if it is correct?

Answer 1

Consider the equation

$$ \sqrt{6x + 27} = 6 + \sqrt{x} $$

The standard way to solve this is to first square both sides:

$$ 6x + 27 = 36 + 12 \sqrt{x} + x $$

isolate $\sqrt{x}$

$$ 12\sqrt{x} = 5x - 9 $$

then square again

$$ 144x = 25x^2 - 90x + 81 $$

then solve the quadratic equation

$$ 0 = 25x^2 - 234 x + 81 = (x-9)(25x-9) $$

Now if we check the solution $x=9$, we get

$$ \sqrt{81} = 6 + \sqrt{9} \qquad \qquad 9 = 6+3 $$

which is true, and if we check $9/25$ we get

$$ \sqrt{\frac{729}{25}} = 6 + \sqrt{\frac{9}{25}} \qquad \qquad \frac{27}{5} = 6 + \frac{3}{5} $$

which is false.