Question

Please solve question 26 and 27​

Answer 1

$\begin{aligned}26.\ \ &\log_{4}\sqrt[3]{8^{\displaystyle\small3+\dfrac{1}{3}\log_2x}}\ =\ 4\\ \\ \Longrightarrow\ \ &\sqrt[3]{8^{\diisplaystyle\small3+\dfrac{1}{3}\log_2x}}\ =\ 4^4\\ \\ \Longrightarrow\ \ &8^{\displaystyle\small3+\dfrac{1}{3}\log_2x}\ =\ \left(4^4\right)^3\ =\ \left(\left(2^2\right)^4\right)^3\ =\ \left(\left(2^3\right)^4\right)^2\ =\ 8^8\end{aligned}$

$\begin{aligned}\Longrightarrow\ \ &3+\dfrac{1}{3}\log_2x\ =\ 8\\ \\ \Longrightarrow\ \ &\dfrac{1}{3}\log_2x=8-3=5\\ \\ \Longrightarrow\ \ &\log_2x=5\times 3=15\\ \\ \Longrightarrow\ \ &\mathbf{x=2^{15}=32768}\end{aligned}$

$\cline{1-}$

$\begin{aligned}27.\ \ &\log_3\left(1+\dfrac{1}{3}\right)+\log_3\left(1+\dfrac{1}{4}\right)+\log_3\left(1+\dfrac{1}{5}\right)+...+\log_3\left(1+\dfrac{1}{80}\right)\\ \\ \Longrightarrow\ \ &\log_3\left(\dfrac{4}{3}\right)+\log_3\left(\dfrac{5}{4}\right)+\log_3\left(\dfrac{6}{5}\right)+...+\log_3\left(\dfrac{81}{80}\right)\\ \\ \Longrightarrow\ \ &\log_3\left(\dfrac{4}{3}\cdot\dfrac{5}{4}\cdot\dfrac{6}{5}\cdot...\cdot\dfrac{80}{79}\cdot\dfrac{81}{80}\right)\end{aligned}$

$\Longrightarrow\ \ \log_3\left(\dfrac{81}{3}\right)\\ \\ \\ \Longrightarrow\ \ \log_327\ \ \Longrightarrow\ \ \log_3\left(3^3\right)\ \ \Longrightarrow\ \ \mathbf{3}$

Answer 2

Answer:

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