Logarithmic Function Simplification: 3^(log_2(4)sqrt(5))/(3^(log_2sqrt(5))

[markl77]

So I am supposed to solve this expression:

3^(log₂(4)sqrt(5))/(3^(log₂sqrt(5))

But I keep getting to dead ends...
can I just cancel out the 3's at the start and simplify each exponent from there? The answer is supposed to be 9 but I keep getting like 38 or something.

 

[JeffM]

So I am supposed to solve this expression:

3^(log₂(4)sqrt(5))/(3^(log₂sqrt(5))

But I keep getting to dead ends...
can I just cancel out the 3's at the start and simplify each exponent from there? The answer is supposed to be 9 but I keep getting like 38 or something.

Is this the expression?

\(\displaystyle \dfrac{3^{log_2(4\sqrt{5})}}{3^{log_2(\sqrt{5})}}\)

\(\displaystyle x = \dfrac{3^{log_2(4\sqrt{5})}}{3^{log_2(\sqrt{5})}} \implies\)

\(\displaystyle log_3(x) = log_3 \left ( \dfrac{3^{log_2(4\sqrt{5})}}{3^{log_2(\sqrt{5})}} \right ) = \)

\(\displaystyle log_3(3^{log_2(4\sqrt{5})}) - log_3(3^{log_2(\sqrt{5})}) = log_2(4\sqrt{5}) * log_3(3) - log_2(\sqrt{5}) * log_3(3) =\)

\(\displaystyle log_2(4\sqrt{5}) * 1 - log_2(\sqrt{5}) * 1 = log_2(4\sqrt{5}) - log_2(\sqrt{5}) =\)

\(\displaystyle log_2(4) + log_2(\sqrt{5}) - log_2(\sqrt{5}) = log_2(4) = 2.\)

In short, \(\displaystyle log_3(x) = 2 \implies x = 3^2 = 9.\)

 

[mmm4444bot]

I am supposed to [simplify] this expression:

3^(log₂(4)*sqrt(5)) / 3^log₂(sqrt(5))

That doesn't equal 9; maybe they typed it incorrectly.

JeffM found a similar expression that equals 9. Here's another way to simplify that one:


\(\displaystyle \dfrac{3^{log_2(4\sqrt{5})}}{3^{log_2(\sqrt{5})}}\)


\(\displaystyle = \dfrac{3^{log_2(4) \;+\; log_2(\sqrt{5})}}{3^{log_2(\sqrt{5})}}\)


\(\displaystyle = \dfrac{3^2 \cdot\; 3^{log_2(\sqrt{5})}}{3^{log_2(\sqrt{5})}}\)


\(\displaystyle = 9\)