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

Study Guide - Practice Questions

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