The task in the image involves using the change-of-base formula to evaluate the expression $\log_{0.7}(11.8)$. Let me guide you through the steps for solving it.

Step 1: Change-of-Base Formula

The change-of-base formula for logarithms is: $\log_b(a) = \frac{\ln(a)}{\ln(b)}$ where $\ln$ is the natural logarithm (logarithm base $e$).

Step 2: Apply the Formula

For the expression $\log_{0.7}(11.8)$, we use the change-of-base formula: $\log_{0.7}(11.8) = \frac{\ln(11.8)}{\ln(0.7)}$

Step 3: Calculate Each Natural Logarithm

1. $\ln(11.8) \approx 2.4681$
2. $\ln(0.7) \approx -0.3567$

Step 4: Division

Now, divide the two results: $\frac{2.4681}{-0.3567} \approx -6.9188$

So, $\log_{0.7}(11.8) \approx -6.9188$.

This value is rounded to four decimal places as required.

Would you like more details on this, or do you have any questions? Here are some related questions you might find helpful:

1. How does the change-of-base formula work in general?
2. What is the significance of the natural logarithm in mathematics?
3. Can you explain why $\ln(0.7)$ is negative?
4. How would the result change if the base were greater than 1?
5. What are some real-life applications of logarithms with bases less than 1?

Tip: When calculating logarithms with bases less than 1, the result is often negative, as seen here.