Answer: The function for estimating the population of wolves in an area is f(x) = 23 + 36 * log(x + 2) where x is the number of years since 2010.
Step-by-step explanation: To write the function in terms of common logarithms, we use the change of base formula:
log_a(b) = log_c(b) / log_c(a)
where a is the new base, b is the original value, and c is the common base (usually 10).
Using this formula, we can rewrite the function as:
f(x) = 23 + 36 * log_10(x + 2)
f(x) = 23 + 36 * (log(x + 2) / log(10))
f(x) = 23 + 36 * (log(x + 2) / 1)
f(x) = 23 + 36 * log(x + 2)
The given function is f(x) = 23 + 36 * log(base e)(x + 2), where x is the number of years since 2010 and f(x) is the estimated number of wolves in the area. Using the change of base formula for logarithms, we can rewrite the function in terms of common logarithms, which is log(base 10). This makes it easier to work with since we are generally more familiar with the base 10 system.
To do this, we first rewrite the formula using the change of base formula for logarithms as f(x) = 23 + 36 * (log(base 10)(x+2) / log(base 10)(e)). Simplifying the right side of the equation, we know that log(base 10)(e) is equal to approximately 0.434. So we can simplify the function further by multiplying the natural logarithm of (x+2) by 36/0.434, which produces the function f(x) = 23 + 82.93 * log(base 10)(x+2).
Finally, we can simplify this expression further by replacing 82.93 with 36/log(base e)(10) = 36/2.303 = 15.624. This gives us the final form of the function in terms of common logarithms: f(x) = 23 + 15.624 * log(base 10)(x+2).
Hope this helps, and have a great day!