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hello i just wanted to check my answer in this test in math :]] i'm not sure if i did everything correctly

hello i just wanted to check my answer in this test in math :]] i'm not sure if i-example-1

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\log _3(\frac{3^{(2)/(3)}}{2^{(1)/(3)}})+\log _(27)(6)\approx1

1) Let's simplify this logarithm expression, making use of properties.


(2)/(3)\log _36-(1)/(3)\log _38+\log _(27)6

2) Let's rewrite that expression turning the factor 2/3 back into an exponent

as well as that -1/3:


\begin{gathered} \log _3\mleft(6^{(2)/(3)}\mright)-\log _3\mleft(8^{(1)/(3)}\mright)+\log _(27)\mleft(6\mright) \\ \end{gathered}

Now, let's rewrite that difference into a quotient:


\begin{gathered} \log _3\mleft(\frac{3^{(2)/(3)}}{2^{(1)/(3)}}\mright)+\log _(27)\mleft(6\mright) \\ 0.4563+0.5436\approx1 \end{gathered}

Since the question does not allow the use of calculator, then we can leave it as the simplest possible expression. Although, the answer is approximately 1.

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