# How do you solve log(x − 3) + log(x − 2) = log(2x + 24)?

May 3, 2018

$x = 9$

#### Explanation:

$\log \left(x - 3\right) + \log \left(x - 2\right) = \log \left(2 x + 24\right)$

$\log \left(\left(x - 3\right) \left(x - 2\right)\right) = \log \left(2 x + 24\right)$

$\left(x - 3\right) \left(x - 2\right) = 2 x + 24$

Simplify the left side:
${x}^{2} - 3 x - 2 x + 6 = 2 x + 24$

${x}^{2} - 5 x + 6 = 2 x + 24$

Add $\textcolor{b l u e}{5 x}$ on both sides of the equation:
${x}^{2} - 5 x + 6 \quad \textcolor{b l u e}{+ \quad 5 x} = 2 x + 24 \quad \textcolor{b l u e}{+ \quad 5 x}$

${x}^{2} + 6 = 7 x + 24$

Move everything to the left side of the equation so that we can factor:
${x}^{2} + 6 - 7 x - 24 = 0$

${x}^{2} - 7 x - 18 = 0$

To factor this, we have to find two numbers that:

• Add up to $- 7$
• Multiply up to $- 18$ from ($1 \cdot - 18$)#

So we have to find a pair of factors of $- 18$ that add up to $- 7$.

We know that the factors of $- 18$ are:
$- 18 , - 9 , - 6 , - 3 , - 2 , - 1 , 1 , 2 , 3 , 6 , 9 , 18$. The two factors that add up to $- 7$ are $- 9 \mathmr{and} 2$.

So the factored form becomes:
$\left(x - 9\right) \left(x + 2\right) = 0$

So
$x - 9 = 0$ and $x + 2 = 0$

$x = 9 \quad$ and $\quad x = - 2$

However, these are not our final solutions! We must check our work when solving log equations by plugging it back into the original equation, since you cannot have a log of $0$ or anything negative.

Let's check our first solution, $x = 9$:
$\log \left(9 - 3\right) + \log \left(9 - 2\right) = \log \left(2 \left(9\right) + 24\right)$

$\log 6 + \log 7 = \log \left(18 + 24\right)$

$\log \left(6 \cdot 7\right) = \log 42$

$\log 42 = \log 42$

This is true! That means that $x = 9$.

Now let's check $x = - 2$:
$\log \left(- 2 - 3\right) + \log \left(- 2 - 2\right) = \log \left(2 \left(- 2\right) + 24\right)$

$\log \left(- 5\right) + \log \left(- 4\right) = \log \left(- 4 + 24\right)$

Oh no! We cannot take the log of a negative number, but $- 5$ and $- 4$ are negative! That means that $x = - 2$ is NOT a solution.

Finally, the answer is $x = 9$.

Hope this helps!