Evaluate :
log 8×log 9log 27\dfrac{\text{log } 8 \times \text{log } 9}{\text{log }27}log 27log 8×log 9
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log 8×log 9log 27⇒log 23×log 32log 33⇒3log 2×2log 33log 3⇒6log 2×log 33log 3⇒6log 23⇒2log2\dfrac{\text{log } 8 \times \text{log } 9}{\text{log }27}\\[1em] \Rightarrow \dfrac{\text{log } 2^3 \times \text{log } 3^2}{\text{log }3^3}\\[1em] \Rightarrow \dfrac{3\text{log } 2 \times 2\text{log } 3}{\text{3log }3}\\[1em] \Rightarrow \dfrac{6\text{log } 2 \times \text{log } 3}{\text{3log }3}\\[1em] \Rightarrow \dfrac{6\text{log } 2}{3}\\[1em] \Rightarrow 2log 2log 27log 8×log 9⇒log 33log 23×log 32⇒3log 33log 2×2log 3⇒3log 36log 2×log 3⇒36log 2⇒2log2
Hence, log 8×log 9log 27=2log2\dfrac{\text{log } 8 \times \text{log } 9}{\text{log }27} = 2 \text{log}2log 27log 8×log 9=2log2.
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