# Question #252ab

Nov 5, 2016

$x = \ln \frac{30}{\ln} \left(24\right) = {\log}_{24} \left(30\right)$

#### Explanation:

${4}^{x} / 5 = {6}^{1 - x}$

Apply the property of exponents that ${a}^{x + y} = {a}^{x} \cdot {a}^{y}$

$\implies {4}^{x} / 5 = {6}^{1} \cdot {6}^{- x}$

Multiply both sides by $5 \cdot {6}^{x}$. Note that ${6}^{- x} \cdot {6}^{x} = {6}^{0} = 1$

$\implies {4}^{x} \cdot {6}^{x} = 30$

Apply the property of exponents that ${a}^{x} \cdot {b}^{x} = {\left(a b\right)}^{x}$

$\implies {\left(4 \cdot 6\right)}^{x} = 30$

$\implies {24}^{x} = 30$

Take a logarithm of both sides

$\implies \ln \left({24}^{x}\right) = \ln \left(30\right)$

Apply the property of logarithms that $\ln \left({a}^{x}\right) = x \ln \left(a\right)$

$\implies x \ln \left(24\right) = \ln \left(30\right)$

$\therefore x = \ln \frac{30}{\ln} \left(24\right) = {\log}_{24} \left(30\right)$