Answer:
n= 2x-6y
Step-by-step explanation:
So we might look at this and have an immediate panic attack, but if there is one thing we might look to to help us, it would be logarithms. Logarithms have many wonderful properties that will help us solve this equation. Let's get started:
1. This is the starting equation. [tex](2^{\frac{1}{2} } )^{n} =\frac{2^x}{8^y}[/tex]
2. Let's use the power to power rule on the left and multiply 1/2 by n:
a. [tex]2^{\frac{1}{2}n } =\frac{2^x}{8^y}[/tex]
3. Next, lets take the log of both sides and use the properties of logarithms:
a. [tex]2^{\frac{1}{2}n } =\frac{2^x}{8^y}[/tex]
b. [tex]log(2^{\frac{1}{2}n }) =log(\frac{2^x}{8^y})[/tex]
c. One property we are allowed to use is bringing down the exponent on the left, and another property we can use is when you have the log of a fraction, that is the same thing as the log(numerator)-log(denominator)
d. [tex]\frac{1}{2}n *log(2) =log(2^x)-log(8^y)[/tex]
e. After we do this, we can use the same property of bringing down the exponent on the right side:
f. [tex]\frac{1}{2}n *log(2) =x*log(2)-y*log(8)[/tex]
g. After this, because log(2) and 1/2 are both numbers and are multiplied by n, we can divide them from both sides, and simplify:
h. [tex]\frac{1}{2}n =\frac{x*log(2)-y*log(8)}{log(2)}[/tex]
i. [tex]2(\frac{1}{2})n =2\frac{x*log(2)-y*log(8)}{log(2)}[/tex]
j. [tex]n =2\frac{x*log(2)-y*log(8)}{log(2)}[/tex]
k. After this, we must simplify the fraction because log(2)/log(2)=1, and log(8)/log(2)=3, these are both because 2^1=2 and 2^3=8 respectively:
l. [tex]n =2(\frac{x*log(2)}{log(2)}-\frac{y*log(8)}{log(2)} )[/tex]
m. Lastly, let's cancel and distribute:
n. [tex]n =2(x-3y)[/tex]
o. [tex]n=2x-6y[/tex]
