# Question #03bcb

Mar 28, 2017

$a = \frac{2 \ln 5 + \ln 2}{5 \ln 5 - 2 \ln 2}$

#### Explanation:

Take the natural logarithm of both sides.
$\ln \left({5}^{5 a - 2}\right) = \ln \left({2}^{2 a + 1}\right)$
Use the log law $\log \left({a}^{b}\right) = b \log \left(a\right)$ to make the index a coefficient.
$\left(5 a - 2\right) \ln \left(5\right) = \left(2 a + 1\right) \ln \left(2\right)$
Expand Brackets
$5 a \ln \left(5\right) - 2 \ln \left(5\right) = 2 a \ln \left(2\right) + \ln \left(2\right)$
Combine like terms on either side of equality
$5 a \ln \left(5\right) - 2 a \ln \left(2\right) = 2 \ln \left(5\right) + \ln \left(2\right)$
Factorise by a on the left hand side
$a \left[5 \ln \left(5\right) - 2 \ln \left(2\right)\right] = 2 \ln \left(5\right) + \ln \left(2\right)$
Divide both sides by $5 \ln \left(5\right) - 2 \ln \left(2\right)$
$\frac{a \left[5 \ln \left(5\right) - 2 \ln \left(2\right)\right]}{5 \ln \left(5\right) - 2 \ln \left(2\right)} = \frac{2 \ln \left(5\right) + \ln \left(2\right)}{5 \ln \left(5\right) - 2 \ln \left(2\right)}$
Cancelling on left hand side leaves;
$a = \frac{2 \ln 5 + \ln 2}{5 \ln 5 - 2 \ln 2}$

Hope that helps :)

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