Let's go through each logarithmic function and apply the change of base formula.

The change of base formula for logarithms is: $\log_a(b) = \frac{\log(b)}{\log(a)}$ where we can use common logarithms (base 10) for evaluation.

Step-by-Step Solutions

1. For $f(x) = \log_9(142)$: $f(142) = \frac{\log(142)}{\log(9)}$ Using a calculator, this evaluates to approximately $f(142) \approx 2.379$.

2. For $g(x) = \log_4(119)$: $g(119) = \frac{\log(119)}{\log(4)}$ Using a calculator, this evaluates to approximately $g(119) \approx 3.438$.

3. For $h(x) = \log_6(51)$: $h(51) = \frac{\log(51)}{\log(6)}$ Using a calculator, this evaluates to approximately $h(51) \approx 2.215$.

4. For $p(t) = \log_7(107)$: $p(107) = \frac{\log(107)}{\log(7)}$ Using a calculator, this evaluates to approximately $p(107) \approx 2.652$.

5. For $f(x) = \log_3(111)$: $f(111) = \frac{\log(111)}{\log(3)}$ Using a calculator, this evaluates to approximately $f(111) \approx 4.853$.

Final Table

Would you like more details on any specific calculation?

Related Questions

1. How is the change of base formula derived?
2. What is the difference between common and natural logarithms?
3. Can logarithmic values be calculated without a calculator?
4. What are some real-world applications of logarithmic functions?
5. How does changing the base of a logarithmic function affect its graph?

Tip: Memorizing the change of base formula can help solve logarithmic problems quickly without needing to know complex bases directly.