# How do you solve 5^(x+3)=6?

Nov 26, 2015

$x = \ln \frac{6}{\ln} \left(5\right) - 3$

#### Explanation:

We will use the property that
$\ln \left({a}^{x}\right) = x \ln \left(a\right)$

${5}^{x + 3} = 6$

$\implies \ln \left({5}^{x + 3}\right) = \ln \left(6\right)$

$\implies \left(x + 3\right) \ln \left(5\right) = \ln \left(6\right)$ $\text{ }$(by the above property)

$\implies x + 3 = \ln \frac{6}{\ln} \left(5\right)$

$\implies x = \ln \frac{6}{\ln} \left(5\right) - 3$

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Note that we can easily check the answer by using the property $\ln \frac{a}{\ln} \left(b\right) = {\log}_{b} \left(a\right)$

Then
${5}^{\ln \frac{6}{\ln \left(5\right)} + 3 - 3} = {5}^{\ln \frac{6}{\ln} \left(5\right)} = {5}^{{\log}_{5} \left(6\right)} = 6$