Express ${log}_{4} a + {log}_{8} (a)^{1 / 3} + [1 / {log}_{a} 8]$ as a logarithm to the base 2.
Give your answer for $a = 4^{9}$

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Solution

${log}_{4} a + {log}_{8} (a)^{1 / 3} + [1 / {log}_{a} 8] = \frac{log a}{log 4} + \frac{2 log a}{9 log 4} + \frac{2 log a}{3 log 4} = {log}_{4} a^{\frac{17}{9}}$
For, $a = 4^{9}$
${log}_{4} a^{\frac{17}{9}} = 17$
Ans: 17

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