To simplify the expression [tex]\(\frac{\log_3 16}{\log_3 0.25}\)[/tex], let's use the properties of logarithms.
Recall the change of base formula which states:
[tex]\[ \log_a b = \frac{\log_c b}{\log_c a} \][/tex]
Here, [tex]\(a = 3\)[/tex] and [tex]\(b\)[/tex] are 16 and 0.25 for each logarithm, respectively.
Step-by-Step Solution:
1. Change the base of the logarithms to base 10:
[tex]\[ \log_3 16 = \frac{\log_{10}(16)}{\log_{10}(3)} \][/tex]
[tex]\[ \log_3 0.25 = \frac{\log_{10}(0.25)}{\log_{10}(3)} \][/tex]
2. Substitute back into the expression:
[tex]\[ \frac{\log_3 16}{\log_3 0.25} = \frac{\frac{\log_{10}(16)}{\log_{10}(3)}}{\frac{\log_{10}(0.25)}{\log_{10}(3)}} \][/tex]
3. Simplify the fraction by canceling [tex]\(\log_{10}(3)\)[/tex]:
[tex]\[ \frac{\log_{10}(16)}{\log_{10}(3)} \div \frac{\log_{10}(0.25)}{\log_{10}(3)} = \frac{\log_{10}(16)}{\log_{10}(0.25)} \][/tex]
4. Recognize that division of logarithms with the same base is equivalent to a logarithm of the division of their arguments:
[tex]\[ \frac{\log_{10}(16)}{\log_{10}(0.25)} = \log_{10}\left(\frac{16}{0.25}\right) \][/tex]
5. Simplify the argument within the logarithm:
[tex]\[ \frac{16}{0.25} = 16 \div 0.25 = 16 \times 4 = 64 \][/tex]
So, we have:
[tex]\[ \log_{10}(64) \][/tex]
6. Since we want the answer in the context of base 3, express [tex]\(64\)[/tex] as a power of 3:
[tex]\[ 64 = 4^3 \][/tex]
Thus:
[tex]\[ \log_{3}(64) = \log_{3}(4^3) \][/tex]
7. Use the power rule for logarithms [tex]\(\log_b (a^c) = c \cdot \log_b a\)[/tex]:
[tex]\[ \log_{3}(4^3) = 3 \cdot \log_{3} 4 \][/tex]
So, the final simplified form of the expression is:
[tex]\[ \frac{\log_3 16}{\log_3 0.25} = \log_{3}(64) = 3 \cdot \log_{3}(4) \][/tex]
Thus, [tex]\(\frac{\log_3 16}{\log_3 0.25} = 3 \cdot \log_{3} 4\)[/tex].
This means that:
[tex]\(\boxed{3}\)[/tex] is the simplified answer if considering powers of 3 without extra logarithmic context.
