Certainly! Let's break down the problem step by step to find the solution for the given logarithmic equation [tex]\( y = \log_{1/4} x \)[/tex].
1. Understanding the problem:
Here, we have a logarithmic equation where [tex]\( y \)[/tex] is the logarithm of [tex]\( x \)[/tex] with base [tex]\( \frac{1}{4} \)[/tex].
2. Equation and Properties:
The logarithmic equation [tex]\( y = \log_{1/4} x \)[/tex] can be rewritten using the definition of logarithms. By definition, [tex]\( y = \log_b a \)[/tex] means [tex]\( b^y = a \)[/tex].
Applying this definition to our equation, we get:
[tex]\[
\left(\frac{1}{4}\right)^y = x
\][/tex]
3. Given Specific Value:
We are given that [tex]\( y \)[/tex] is 2. Substituting this value into our equation, we have:
[tex]\[
\left(\frac{1}{4}\right)^2 = x
\][/tex]
4. Simplifying the Expression:
Next, let's simplify the expression [tex]\( \left(\frac{1}{4}\right)^2 \)[/tex]:
[tex]\[
\left(\frac{1}{4}\right)^2 = \frac{1}{4} \times \frac{1}{4} = \frac{1}{16}
\][/tex]
Therefore, when [tex]\( y = 2 \)[/tex], the value of [tex]\( x \)[/tex] such that [tex]\( y = \log_{1/4} x \)[/tex] is:
[tex]\[
x = \frac{1}{16}
\][/tex]
5. Summarizing the Results:
The base of our logarithm is [tex]\( \frac{1}{4} \)[/tex], and when [tex]\( y = 2 \)[/tex], the corresponding [tex]\( x \)[/tex] value is [tex]\( \frac{1}{16} \)[/tex].
Thus, the detailed solution can be summarized as:
- Given the logarithmic equation [tex]\( y = \log_{1/4} x \)[/tex],
- When [tex]\( y = 2 \)[/tex],
- The base of the logarithm is [tex]\( \frac{1}{4} \)[/tex],
- Therefore, [tex]\( x \)[/tex] is [tex]\( \frac{1}{16} \)[/tex].
Concluding, the calculated (logarithmic) relationship yields us the results [tex]\( y = 2 \)[/tex], base = [tex]\( \frac{1}{4} \)[/tex], and [tex]\( x = 0.0625 \)[/tex].
