To evaluate the expression $\log_{12} 89.5$, you can use the change of base formula, which states:

$\log_{a} b = \frac{\log b}{\log a}$

Thus,

$\log_{12} 89.5 = \frac{\log 89.5}{\log 12}$

Now, let's calculate this.

1. Calculate $\log 89.5$: $\log 89.5 \approx 1.9525$

2. Calculate $\log 12$: $\log 12 \approx 1.0792$

3. Divide these values: $\log_{12} 89.5 \approx \frac{1.9525}{1.0792} \approx 1.8095$

Therefore, the value of $\log_{12} 89.5$ is approximately 1.8095 (rounded to four decimal places).

Would you like a further breakdown of any steps?

Here are some related questions to explore this concept further:

1. How would you calculate $\log_5 125$ using the change of base formula?
2. What is the value of $\log_{12} 144$?
3. How do logarithmic properties simplify calculations?
4. Can you rewrite $\log_{12} (89.5 \times 144)$ in terms of separate logarithms?
5. How do you interpret a logarithm like $\log_{12} 89.5$ graphically?

Tip: When calculating logarithms of non-standard bases, the change of base formula can be a quick tool with a standard calculator that only has common (base-10) or natural (base-e) logarithms.