How do you solve log_7x+log_7(x+5)=log_7(14)?

Dec 8, 2016

$x = - 7$

$x = + 2$

Explanation:

Notice is that everything is to logs of the same base. This means that we can get rid of the logs.

Note that $\log \left(a\right) + \log \left(b\right)$ is the same thing as $\log \left(a \times b\right)$

This gives us:

${\log}_{7} \left(x \left(x + 5\right)\right) = {\log}_{7} \left(14\right)$

$\implies x \left(x + 5\right) = 14$

${x}^{2} + 5 x - 14 = 0$
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Note that $2 \times 7 = 14 \text{ and } 7 - 2 = 5$

$\left(x + 7\right) \left(x - 2\right) = 0$

$x = - 7$

$x = + 2$