Question

f(0) = -\log_{3} 37 + 1\
f(0) = -\frac{\ln 37}{\ln 3} + 1\
f(0) \approx -2.2870

Explanation:

Step1: Recall Change of Base Formula

The change of base formula for logarithms is $\log_{a}b=\frac{\ln b}{\ln a}$. So, $\log_{3}37 = \frac{\ln 37}{\ln 3}$.

Step2: Substitute into the Function

We have $f(0)=-\log_{3}37 + 1$. Substituting the change of base formula result, we get $f(0)=-\frac{\ln 37}{\ln 3}+1$.

Step3: Calculate the Numerical Value

First, calculate $\ln 37\approx 3.6109$, $\ln 3\approx 1.0986$. Then $\frac{\ln 37}{\ln 3}\approx\frac{3.6109}{1.0986}\approx 3.2867$. Then $-\frac{\ln 37}{\ln 3}+1\approx - 3.2867 + 1=-2.2867\approx - 2.2870$.

Answer:

$f(0)\approx - 2.2870$ (The process shows how to convert the logarithm with base 3 to natural logarithm and then compute the numerical value to get the approximation of $f(0)$.)