Solve the following equation find x
${log}_{4} (2 {log}_{3} (1 + {log}_{2} (1 + 3 {log}_{3} x))) = \frac{1}{2}$

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Solution

${log}_{4} (2 {log}_{3} (1 + {log}_{2} (1 + 3 {log}_{3} x))) = \frac{1}{2}$
using the logarithm property :
${log}_{b} a = k \Rightarrow a = b^{k} ∴ 2 {log}_{3} (1 + {log}_{2} (1 + 3 {log}_{3} x)) = 4^{\frac{1}{2}} {log}_{3} (1 + {log}_{2} (1 + 3 {log}_{3} x)) = 1 1 + {log}_{2} (1 + 3 {log}_{3} x) = 3^{1} {log}_{2} (1 + 3 {log}_{3} x) = 2 1 + 3 {log}_{3} x = 2^{2} 3 {log}_{3} x = 3 {log}_{3} x = 1 x = 3$

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