Question 1

Given the complex numbers $$z = -1 + 3i$$ and $$w = 5 - 3i$$, what is the value of $$z + w$$?

Accepted Answer

$$4 + 0i$$

Question 2

What is the product of $$z$$ and its conjugate, where $$z = -1 + 3i$$?

Accepted Answer

$$10$$

Question 3

Write the exponential equation $$3^x = 12 in $$logarithmic form and solve for $$x.$$

Accepted Answer

$$x = \log_3$ 12$$

Question 4

Express $$\log_5$ x = 3 in $$exponential form and solve for $$x.$$

Accepted Answer

$$x = 125$$

Question 5

Simplify the expression $$\log 75 + \log 2 - \log 10 as a $$single logarithm.

Accepted Answer

$$\log 15$$

Question 6

Write $$\log(x^2$ - 6x + 5) - \log(x - 5) + \log(x + 4) as a $$single logarithm and simplify.

Accepted Answer

$$\log(x + 4)$$

Question 7

Express $$\log(x^3$ y^2$ / z^5$) as a $$sum and difference of simpler logarithms.

Accepted Answer

$$3 \log x + 2 \log y - 5 \log z$$

Question 8

Solve for $$x$$: $$2^x = 5$$

Accepted Answer

$$x = \log_2$ 5$$

Question 9

Solve for $$x$$: $$3^x = 360$$

Accepted Answer

$$x = \log_3$ 360$$

Question 10

Solve for $$x$$: $$\log_7$(x + 4) = \log_7$(2x - 1)$$

Accepted Answer

$$x = 5$$

Question 11

Solve for $$x$$: $$\log_2$(x + 4) + \log_2$(x) = 1$$

Accepted Answer

$$x = 1$$

Question 12

A medical test result increases exponentially according to $$N = N_0$ e$$^{kt}$$. If the initial value is N_0$ = 100$$, $$k = 0.2$$, and $$t = 5$$, what is $$N$$?

Accepted Answer

$$N = 100 e$$^{1}$$

Question 13

If the pH of a solution is defined as $$pH = -\log_{10}[H^+]$$, what is the hydrogen ion concentration $$[H^+]$$ when $$pH = 3$$?

Accepted Answer

$$[H^+] = 10$$^{-3}$$

Question 14

A population doubles every 6 years. If the initial population is $$P_0$$, what is the population after 18 years?

Accepted Answer

$$P = P_0$ 	imes 2^3$

Question 15

Which of the following is the correct logarithmic form of the exponential equation $$47 = 267^x$$?

Accepted Answer

$$x = \log_{267} 47$$

Question 16

Express $$\log_{10} 125 = 3 in $$exponential form and solve for $$125.$$

Accepted Answer

$$10^3 = 125$

Question 17

Given $$\log_2$ x = 0.5$$, solve for $$x.$$

Accepted Answer

$$x = \sqrt{2}$$

Question 18

If $$\log_a b = c$$, which of the following is the correct exponential form?

Accepted Answer

$$a^c = b$$

Question 19

A radioactive substance decays according to $$N = N_0$ e$$^{-kt}$$. If N_0$ = 500$$, $$k = 0.1$$, and $$t = 10$$, what is $$N$$?

Accepted Answer

$$N = 500 e$$^{-1}$$

College Algebra: Complex Numbers, Exponents, and Logarithms – Exam Review Study Notes

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