Q: The population of a city grows exponentially at a rate of $$2.14\%$$ per year. If the initial population is $$6.18$$ million, which function models the population $$P(t)$$ after $$t$$ years?

$$P(t) = 6.18 \times e$$^{0.0214t}$$

Question 1

Find the inverse function of $$f(x) = \dfrac{2x}{3x-1}. $$Which of the following is correct?

Accepted Answer

Question 2

Evaluate $$\log_4$ 2.$$

Accepted Answer

$$\dfrac{1}{2}$$

Question 3

Convert the exponential equation $$8 = 2^x to a $$logarithmic equation.

Accepted Answer

$$x = \log_2$ 8$$

Question 4

Express $$\log_a \dfrac{b^5$ a^3$}{c^4$} in $$terms of sums and differences of logarithms.

Accepted Answer

$$5\log_a b + 3\log_a a - 4\log_a c$$

Question 5

Given $$\log_a 2 = A$$ and $$\log_a 3 = B$$, find $$\log_a 36.$$

Accepted Answer

$$2A + 2B$$

Question 6

Simplify $$\ln e$$^{-3}$$.

Accepted Answer

$$-3$$

Question 7

Solve for $$x$$: $$\log_2$ (x+2) + \log_2$ (x-2) = 2.$$

Accepted Answer

$$x = 4$$

Question 8

The population of a city grows exponentially at a rate of $$2.14\%$$ per year. If the initial population is $$6.18$$ million, which function models the population $$P(t)$$ after $$t$$ years?

Accepted Answer

$$P(t) = 6.18 	imes e$$^{0.0214t}$$

Question 9

Find the zeros and their multiplicities for $$f(x) = (x^2$ - 5x + 6)^2$ (x^2$ - 4). $$Which of the following is correct?

Accepted Answer

Question 10

Solve the inequality $$x^2 - 8$$ \u003e 6x$$.

Accepted Answer

$$x  6$$

Question 11

Solve for $$x$$: $$32^{x+3} = 9$.

Accepted Answer

$$x = -2.5$$

Question 12

Express $$3\log_b x + \dfrac{1}{2}\log_b y - 2\log_b z as a $$single logarithm.

Accepted Answer

$$\log_b \dfrac{x^3$ \sqrt{y}}{z^2$}$$

Question 13

Solve the system: $$\begin{cases} 2x + y = 7 \ x - 5y = 4 \end{cases}$$

Accepted Answer

Question 14

Solve the logarithmic equation: $$\log x + \log (x+4) = \log 12.$$

Accepted Answer

$$x = 2$$

Question 15

Which of the following best describes the transformation of the graph of $$f(x) = 5 - \log(x+2)$$ compared to the basic logarithmic function $$y = \log x$$?

Accepted Answer

Shift left 2 units, reflect over the x-axis, and shift up 5 units

Question 16

Solve for $$x$$: $$\log_{27} 3 = x.$$

Accepted Answer

$$x = \dfrac{1}{3}$$

Question 17

If $$\log_5$ 53 = x$$, what is the value of $$x$$?

Accepted Answer

$$x = 3$$

Question 18

Solve the system of equations: $$\begin{cases} x + 6y + 3z = 4 \ 2x + y + 2z = 3 \ 3x - 2y + z = 0 \end{cases}$$

Accepted Answer

Question 19

Find the doubling time for the population growth function $$P(t) = 6.18 e$$^{0.0214t}$$.

Accepted Answer

$$\dfrac{\ln 2}{0.0214}$$

Question 20

Simplify $$e^{\ln 3k}.$$

Accepted Answer

$$3k$$

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