# How do you solve log_a (X+4) - log_a (X-7) = log_a (X-8) - log_a (X-9)?

Jul 24, 2015

$X = \frac{46}{5} = 9.2$

#### Explanation:

First, use the property that ${\log}_{a} \left(\frac{C}{D}\right) = {\log}_{a} \left(C\right) - {\log}_{a} \left(D\right)$ to rewrite the equation as

${\log}_{a} \left(\frac{X + 4}{X - 7}\right) = {\log}_{a} \left(\frac{X - 8}{X - 9}\right)$

Next, use the fact that ${\log}_{a} \left(z\right)$ is a one-to-one function (when $a > 0$ and $a \ne 1$) to say that the inputs are the same:

$\frac{X + 4}{X - 7} = \frac{X - 8}{X - 9}$

Now cross-multiply to get:

$\left(X + 4\right) \left(X - 9\right) = \left(X - 8\right) \left(X - 7\right)$

After using FOIL, this becomes

${X}^{2} - 5 X - 36 = {X}^{2} - 15 X + 56$.

Rearranging and canceling appropriately gives

$10 X = 92$ so that $X = \frac{92}{10} = \frac{46}{5} = 9.2$

You should always check your answer in the original equation. When $X = 9.2$, we have

$X + 4 = 13.2$, $X - 7 = 2.2$, $X - 8 = 1.2$, and $X - 9 = 0.2$

You can use any value for $a > 0$ except $a = 1$. For instance, if we use $a = 10$, then:

${\log}_{10} \left(13.2\right) \approx 1.120574$, ${\log}_{10} \left(2.2\right) \approx 0.342423$, ${\log}_{10} \left(1.2\right) \approx 0.079181$, and ${\log}_{10} \left(0.2\right) \approx - 0.698970$

and ${\log}_{10} \left(13.2\right) - {\log}_{10} \left(2.2\right) \approx 1.120574 - 0.342423 = 0.778151$ and ${\log}_{10} \left(1.2\right) - {\log}_{10} \left(0.2\right) \approx 0.079181 - \left(- 0.698970\right) = 0.778151$

Jul 24, 2015

I found $x = 9.2$

#### Explanation:

You can use the fact that: ${\log}_{a} x - {\log}_{a} y = {\log}_{a} \left(\frac{x}{y}\right)$ to get:
${\log}_{a} \left(\frac{x + 4}{x - 7}\right) = {\log}_{a} \left(\frac{x - 8}{x - 9}\right)$
take the power of $a$ on bothe sides to get rid of the logs:
${\cancel{a}}^{\cancel{{\log}_{a}} \left(\frac{x + 4}{x - 7}\right)} = {\cancel{a}}^{\cancel{{\log}_{a}} \left(\frac{x - 8}{x - 9}\right)}$
$\frac{x + 4}{x - 7} = \frac{x - 8}{x - 9}$
$\left(x + 4\right) \left(x - 9\right) = \left(x - 8\right) \left(x - 7\right)$
$\cancel{{x}^{2}} - 9 x + 4 x - 36 = \cancel{{x}^{2}} - 7 x - 8 x + 56$
$10 x = 92$
$x = 9.2$