Question 1

Let $$f(x) = 10 + \ln(x^2$ + 49). $$What is the domain of $$f(x)$$?

Accepted Answer

$$(-\infty, \infty)$$

Question 2

Find the range of $$g(x) = e$$^{-x}$$ + B$$, where $$A \u003e B \u003e 0$$ are real numbers.

Accepted Answer

$$(B, \infty)$$

Question 3

Expand and simplify completely: $$\log_2$(16x/40).$$

Accepted Answer

$$\log_2$(16x) - \log_2$(40)$$

Question 4

Identify an equivalent expression to $$\log_5$(xy).$$

Accepted Answer

$$\log_5$ x + \log_5$ y$$

Question 5

Evaluate $$\log_2$ 256 - \log_2$ 2.$$

Accepted Answer

$$6$$

Question 6

If interest on savings is compounded continuously at 15% annually, how long will it take for the accumulated amount to triple? (Approximate to 1 decimal place.)

Accepted Answer

$$7.8$$ years

Question 7

A certificate of deposit purchased today will be worth $7200 after five years from now. Interest is 11% compounded continuously. What amount is deposited?

Accepted Answer

$$S = 7200 e$$^{-0.55}$$

Question 8

Determine the effective rate, $$r_e$$, for a loan obtained at nominal interest rate of 12% compounded quarterly.

Accepted Answer

$$r_e = (1 + \frac{0.12}{4})^4$ - 1$$

Question 9

Assume $$A, B, C, D$$ are nonzero constants. Find the inverse of the one-to-one function $$f(x) = Ae$$^{Bx+C}$$ - D.$$

Accepted Answer

$$f^{-1}(x$$) = \frac{1}{B} \ln\left(\frac{x+D}{A}\right) - \frac{C}{B}$$

Question 10

Express as a single logarithm and simplify as much as possible: $$\log_3$(2041(x+4)y^2$$) - \$$log_7((x+y)^4$) + \frac{3}{2} \log_2$(z+u).$$

Accepted Answer

$$\log_3$(2041) + \log_3$(x+4) + 2\log_3$(y) - 4\log_7$(x+y) + \frac{3}{2}\log_2$(z+u)$$

Question 11

Solve $$e^{2x} = 3$ for x$.

Accepted Answer

$$x = \frac{1}{2} \ln 3$$

Question 12

Given $$\log_2$(x+3) = 2 - \log_2$(x+6)$$, solve for $$x.$$

Accepted Answer

$$x = 2$$

Question 13

Let $$f(x) = e$$^{10\log x^2 + 24}$$ and g(x$$) = $$e^{2x+24}. $$Evaluate $$f(2) / g(2).$$

Accepted Answer

$$e^{8}$$

Question 14

Aaron saved money on a fixed deposit account offering annual interest rate 12.5%. After 7 years, the savings accumulated to $$45,500. $$Determine the principal amount given that interest was compounded annually.

Accepted Answer

$$P = \frac{45500}{(1.125)^7$}$$

Question 15

How long will it take the car purchased at $$32,500 to $$drop to $$2,500 if $$the depreciation rate is $$9.25\%$$? Use the model $$S = S_0$ e$$^{-rt}$$.

Accepted Answer

$$t = \frac{\ln(2500/32500)}{-0.0925}$$

Question 16

Find the sixteenth term in the arithmetic sequence $$-10, -6, -2, ....$$

Accepted Answer

$$a_{16} = -10 + 15 	imes 4$$

Question 17

For $$k a $$positive integer, find the general term $$a_k$$ for the arithmetic sequence $$-10, -6, -2, ....$$

Accepted Answer

$$a_k = -10 + (k-1) 	imes 4$$

Question 18

Find the sum of the first eight terms of the arithmetic sequence $$-10, -6, -2, .... $$Approximate to two decimal places.

Accepted Answer

$$S_8$$ = \frac{8}{2}(-10 + 18)$$

Question 19

Evaluate, if possible, $$\sum_{k=1}^8 a_k$$ for $$a_k = -10 + (k-1) 	imes 4.$$

Accepted Answer

$$\sum_{k=1}^8 a_k = 32$$

Question 20

The sum of an infinite number of terms of a convergent geometric sequence with $$|r| \u003c 1 is $$given by:

Accepted Answer

$$S = \frac{a}{1 - r}$$

Precalculus Study Guide: Logarithms, Exponentials, Sequences, and Financial Applications

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