If `log_(b)x=0.2, and log_(b)y=0.4,` what is the relationship between x and y ?

Options

A. y=2x
B. `y=x^(2)`
C. `y=sqrt(x)`
D. `xy=0.6`

Explanation

To find the relationship between \( x \) and \( y \) given that \( \log_b x = 0.2 \) and \( \log_b y = 0.4 \), we can follow these steps:

### Step 1: Convert logarithmic equations to exponential form
Using the property of logarithms that states if \( \log_b a = c \), then \( a = b^c \):

1. From \( \log_b x = 0.2 \), we can write:
\[ x = b^{0.2} \]

2. From \( \log_b y = 0.4 \), we can write:
\[ y = b^{0.4} \]

### Step 2: Express \( y \) in terms of \( x \)
Now, we can express \( y \) in terms of \( x \). We know that:
\[ y = b^{0.4} \]
We can rewrite \( 0.4 \) as \( 2 \times 0.2 \):
\[ y = b^{2 \times 0.2} = (b^{0.2})^2 \]

### Step 3: Substitute \( x \) into the equation
Since we have already established that \( x = b^{0.2} \), we can substitute \( x \) into the equation:
\[ y = (b^{0.2})^2 = x^2 \]

### Conclusion
Thus, we find the relationship between \( x \) and \( y \):
\[ y = x^2 \]