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Let b > 1. Find the value of 

\(\log_{b} \left(\dfrac 2 3\right) + \log_{b} \left(\dfrac 3 2\right).\)

 Apr 25, 2023
 #1
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\(\log_{b} \left(\dfrac 2 3\right) + \log_{b} \left(\dfrac 3 2\right) = 1\)

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 Apr 25, 2023
 #2
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Let \(x = \log_b ({2 \over 3})\) and \(y = \log_b ({2 \over 3})\)

 

Then \(b^x = {2 \over 3}\) and \(b^y = {3 \over 2}\)

 

Multiplying the two gives us \(b^x \times b^y = b^{x + y} = 1\)

 

Now, notice that the only way the expression equals 1 is if \(x + y = \color{brown}\boxed{0}\).

 

Note: In general, \(\log_b({x}) + \log_b({y}) = \log_b({xy})\)

 Apr 26, 2023

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