The formula $$ t=\frac{1}{c}[\ln A-\ln (A-N)] $$ describes the time, $$ t $$, in weeks, that it takes to achieve mastery of a portion of a task, where $$ A $$ is the maximum leaming possible, $$ N $$ is the portion of the learming that is to be achieved, and c is a constant used to measure an individual's leaming style. a. Express the formula so that the expression in brackets is written as a single logarithm. b. Use the form of the formula from part (a) to answer this question. An animal learning sign language can master a maximum of 55 signs. How many weeks will it take the animal to master 30 signs if the constant c for that animal is 0.02 ? a. Express the formula in the brackets as a single logarithm. $$ t=\square $$ (Use parentheses to clearly denote the argument of each function.)

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**a.**  
Starting with  
$$
t = \frac{1}{c} \left[\ln A - \ln (A-N)\right],
$$  
use the logarithm property  
$$
\ln a - \ln b = \ln \left(\frac{a}{b}\right)
$$  
to combine the logarithms:  
$$
\ln A - \ln (A-N) = \ln\left(\frac{A}{A-N}\right).
$$  
Thus, the formula becomes:  
$$
t = \frac{1}{c}\ln\left(\frac{A}{A-N}\right).
$$

**b.**  
For an animal that can master a maximum of $$ A = 55 $$ signs, to master $$ N = 30 $$ signs with a constant $$ c = 0.02 $$, substitute these values into the formula:  
$$
t = \frac{1}{0.02}\ln\left(\frac{55}{55-30}\right).
$$  
Calculate the expression inside the logarithm:  
$$
55 - 30 = 25, \quad \text{so} \quad \frac{55}{25} = \frac{11}{5} = 2.2.
$$  
Thus,  
$$
t = \frac{1}{0.02}\ln(2.2).
$$  
Since $$\frac{1}{0.02} = 50$$, we have:  
$$
t = 50\ln(2.2).
$$  
Approximating $$\ln(2.2) \approx 0.7885$$,  
$$
t \approx 50 \times 0.7885 = 39.425 \text{ weeks}.
$$

So, the animal will take approximately $$ \boxed{39.43 \text{ weeks}} $$ to master 30 signs.
**a.** Combine the logarithms in the formula: $$ t = \frac{1}{c} \ln\left(\frac{A}{A-N}\right). $$ **b.** Given $$ A = 55 $$, $$ N = 30 $$, and $$ c = 0.02 $$: $$ t = \frac{1}{0.02} \ln\left(\frac{55}{25}\right) = 50 \ln(2.2) \approx 39.43 \text{ weeks}. $$

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