To solve the exponential equation
[tex]\[ 4 \cdot 4^y = \left( \frac{\sqrt{2}}{4} \right)^y, \][/tex]
follow these steps:
### Step 1: Simplify each side
First, rewrite the equation for clarity:
[tex]\[ 4 \cdot 4^y = \left( \frac{\sqrt{2}}{4} \right)^y.\][/tex]
### Step 2: Express 4 and [tex]\(\sqrt{2}\)[/tex] in terms of base 2
Notice that [tex]\(4 = 2^2\)[/tex] and [tex]\(\sqrt{2} = 2^{1/2}\)[/tex]. We can rewrite [tex]\(4^y\)[/tex] and [tex]\(\left( \frac{\sqrt{2}}{4} \right)^y\)[/tex] using these expressions:
[tex]\[ 4 \cdot (2^2)^y = \left( \frac{2^{1/2}}{2^2} \right)^y.\][/tex]
### Step 3: Simplify the exponents
[tex]\[ 4 \cdot 2^{2y} = \left( 2^{1/2 - 2} \right)^y.\][/tex]
[tex]\[ 4 \cdot 2^{2y} = 2^{y(1/2 - 2)}.\][/tex]
### Step 4: Combine the terms involving powers of 2
[tex]\[4 \cdot 2^{2y} = 2^{y(1/2 - 2)}.\][/tex]
### Step 5: Write 4 as a power of 2
Since [tex]\(4 = 2^2\)[/tex], we substitute it into the equation:
[tex]\[2^2 \cdot 2^{2y} = 2^{(-3/2)y}.\][/tex]
### Step 6: Combine exponents on the left-hand side
[tex]\[2^{2 + 2y} = 2^{(-3/2)y}.\][/tex]
### Step 7: Set the exponents equal to each other
Since the bases are the same, we set their exponents equal:
[tex]\[2 + 2y = -\frac{3}{2}y.\][/tex]
### Step 8: Solve for [tex]\(y\)[/tex]
Rearrange the equation to solve for [tex]\(y\)[/tex]:
[tex]\[2 + 2y = -\frac{3}{2}y.\][/tex]
Multiply every term by 2 to clear the fraction:
[tex]\[4 + 4y = -3y.\][/tex]
Combine like terms:
[tex]\[4 + 4y + 3y = 0.\][/tex]
[tex]\[4 + 7y = 0.\][/tex]
Subtract 4 from both sides:
[tex]\[7y = -4.\][/tex]
Divide both sides by 7:
[tex]\[ y = -\frac{4}{7} \approx -0.571428571428572.\][/tex]
So, the solution to the equation
[tex]\[4 \cdot 4^y = \left( \frac{\sqrt{2}}{4} \right)^y\][/tex]
is
[tex]\[ y \approx -0.571428571428572.\][/tex]
