(a) Write the expression in terms of natural logarithms. (Express all logarithmic functions in terms of ln(x) in your answer.)
log_(8.5) \(127\) =
Give a calculator approximation (correct to four decimal places).
log_(8.5) \(127\) ~=
To express the given logarithm in terms of natural logarithms, we can use the change of base formula. The change of base formula states that for any positive numbers \(a\) and \(b\), and any positive base \(c\), the following equation holds:
log_c \(b\) = log_a \(b\) / log_a \(c\)
In this case, we want to express log_(8.5) \(127\) in terms of ln(x). So, applying the change of base formula, we have:
log_(8.5) \(127\) = log \(127\) / log \(8.5\)
Now, to find the calculator approximation:
1. Calculate the logarithm of 127 to the base 10: log \(127\).
2. Calculate the logarithm of 8.5 to the base 10: log \(8.5\).
3. Divide the result from step 1 by the result from step 2.
Using a calculator, log \(127\) ≈ 2.1038 and log \(8.5\) ≈ 0.9294.
Therefore, log_(8.5) \(127\) ≈ 2.1038 / 0.9294.
Evaluating this expression gives us approximately 2.2630 (rounded to four decimal places).
So, log_(8.5) \(127\) ~= 2.2630