Question

. Write each number in terms of natural logarithms, and then use the properties of logarithms to show that it is a

Answer 1

Answer:

The answer is 3/4

Step-by-step explanation:

Hi, we need to change the base of the logarithm, for that we need to use the following formula.

$Log_{b} (a)=\frac{Ln(a)}{Ln(b)}$

In our case, this is:

$Log_{9} (\sqrt{27} )=\frac{Ln(\sqrt{27} )}{Ln(9)}$

Which is the same as:

$\frac{Ln(27)^{\frac{1}{2} }}{Ln(9)}$

Now, let´s solve this using the log properties

$\frac{1}{2} (\frac{Ln(27)}{Ln(9)} )$

$\frac{1}{2} (\frac{Ln(9*3)}{Ln(9)} )$

$\frac{1}{2} (\frac{Ln(9)+Ln(3)}{Ln(9)} )$

$\frac{1}{2} (\frac{Ln(9)}{Ln(9)}+\frac{Ln(3)}{Ln(9)} )$

We can change 3 for 9^(1/2)

$\frac{1}{2} (\frac{Ln(9)}{Ln(9)}+\frac{Ln(9^{\frac{1}{2} } )}{Ln(9)} )$

$\frac{1}{2} (\frac{Ln(9)}{Ln(9)}+\frac{Ln(9 )}{2*Ln(9)} )$

Since Ln(9) / Ln(9) =1, we get.

$\frac{1}{2} (1+\frac{1}{2} )$

$\frac{1}{2} (\frac{3}{2} )=\frac{3}{4}$

Best of luck.