When combining logarithms there are a few rules to remember. First, addition outside a log is multiplication inside. Subtraction outside a log is division inside. Finally, multiplication/division outside are exponents inside. Also, ln(1)=0. So,

turns into 

and then 

Recall the standard form of an exponential equation: 

Plug in the two ordered pairs:

Solve for a in the first equation:

Now plug this value into the second equation:

Simplify:

Multiply both sides by 3/2 and simplify:

Plug this into the first equation to solve for a:

Plug the values of a and b into the general form to get the answer:

The standard formula for population growth is

,

where  is the population after time t,  is the initial population, and r is the rate of growth as a decimal, per unit time. (In other words, r and t must have the same units.)

The problem also gives us initial population as 5000 and growth rate as 5% or .05.

Plug in:

Simplify the parentheses:

Evaluate:

Round to get the final answer:

We recall that the formula for population decay is

,

where  is the population at time t,  is the initial population, and r is the rate of decrease per unit time (same unit as t).

 is half of , so we can write 

.

Simplify and eliminate common factors:

Take the log of both sides. Note that the log has base .98.

Use the change of base theorem to rewrite the log:

Round:

 the -int will be when  and the -int will not exist because all the values of this function are positive. 

So the curve must be D since that is the only curve that intersects at the point 

This sum can be determined using the formula for the sum of an infinite geometric series, with initial term  and common ratio :

An arithmetic sequence is one in which there is a common difference between consecutive terms. For example, the sequence {2, 5, 8, 11} is an arithmetic sequence, because each term can be found by adding three to the term before it. 

Let  denote the nth term of the sequence. Then the following formula can be used for arithmetic sequences in general:

, where d is the common difference between two consecutive terms. 

We are given the 4th and 8th terms in the sequence, so we can write the following equations:

.

We now have a system of two equations with two unknowns:

Let us solve this system by subtracting the equation  from the equation . The result of this subtraction is

.

This means that d = 2.5.

Using the equation , we can find the first term of the sequence.

Ultimately, we are asked to find the hundredth term of the sequence.

The answer is 220.

The sum of all odd numbers is another way to construct perfect squares.  To see why this is, we can construct the series as follows.

We draw  from the series by subtracting one from each term.

We discard the 0 term and factor 2 out of the remaining terms.

And finally we use the property that  to evaluate the series.  

The smallest value of  where the square exceeds 200 is .

The terms of an arithmetic series are generated by the relation

,

where is the 1st term, is the nth term, and d is the common difference.

For

,

for 

.

The first step is to find .

, so

.

Now to find , when .

Use the generating relation

.