To evaluate [tex]\(\log_4 0.25\)[/tex], we proceed as follows:
1. Understand the logarithm:
[tex]\(\log_b a\)[/tex] represents the power to which the base [tex]\(b\)[/tex] must be raised to obtain the number [tex]\(a\)[/tex]. So, we need to find the exponent [tex]\(x\)[/tex] such that [tex]\(4^x = 0.25\)[/tex].
2. Express 0.25 as a power of 4:
Notice that [tex]\(0.25 = \frac{1}{4}\)[/tex]. This can be written in terms of exponents of 4:
[tex]\[
0.25 = 4^{-1}
\][/tex]
3. Set up the equation:
We have:
[tex]\[
\log_4 0.25 = \log_4 4^{-1}
\][/tex]
According to the property of logarithms:
[tex]\[
\log_b b^x = x
\][/tex]
Therefore:
[tex]\[
\log_4 4^{-1} = -1
\][/tex]
However, considering the more thorough analytical approach commonly used with algebraic manipulation and knowing the precise numerical method provided, we can refine this understanding by recognizing logarithmic properties and the exact result. The answer then includes evaluating using natural logarithms:
4. Using the change of base formula:
Recall the change of base formula for logarithms:
[tex]\[
\log_b a = \frac{\log_c a}{\log_c b}
\][/tex]
Choosing natural logarithm (base [tex]\(e\)[/tex]), we get:
[tex]\[
\log_4 \left(\frac{1}{4}\right) = \frac{\log \left(\frac{1}{4}\right)}{\log 4}
\][/tex]
5. Apply the logarithm properties:
Using properties of logarithms:
[tex]\[
\log \left(\frac{1}{4}\right) = \log 4^{-1} = -\log 4
\][/tex]
Substitute this into our expression:
[tex]\[
\log_4 \left(\frac{1}{4}\right) = \frac{-\log 4}{\log 4} = -1
\][/tex]
Since we arrive conclusively that [tex]\(\log_4 (0.25) = -1.38629436111989 / \log(4)\)[/tex], the detailed answer confirms the numerical result evaluated:
Thus, the step-by-step solution is:
[tex]\[
\log_4 0.25 = -1.38629436111989 / \log(4)
\][/tex] and confirms [tex]\(\log_4 0.25 = -1\)[/tex].
