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

Complex Numbers

Definition and Basic Operations

Complex numbers are numbers of the form z = a + bi, where a and b are real numbers, and i is the imaginary unit with the property that $i^2 = -1$.

• Real Part: The real part of z is a.

• Imaginary Part: The imaginary part of z is b.

• Complex Conjugate: The conjugate of z = a + bi is \overline{z} = a - bi.

• Multiplying by the Conjugate: $z \cdot \overline{z} = (a + bi)(a - bi) = a^2 + b^2$ (always a non-negative real number).

Example: If $z = -1 + 3i$, then $\overline{z} = -1 - 3i$ and $z \cdot \overline{z} = (-1)^2 + (3)^2 = 1 + 9 = 10$.

Exponential and Logarithmic Equations

Exponential Form and Logarithmic Form

Exponential and logarithmic expressions are inverse operations. Understanding how to convert between them is essential in solving equations.

• Exponential Form: $a^x = b$

• Logarithmic Form: $\log_a b = x$

Example: $2^3 = 8$ is equivalent to $\log_2 8 = 3$.

Writing Expressions in Logarithmic Form and Solving for x

• Given: $a^x = b$

• Logarithmic Form: $\log_a b = x$

• Solving for x: Take the logarithm of both sides if necessary.

Examples:

• $3^x = 12$ → $x = \log_3 12$

• $4^x = 256$ → $x = \log_4 256$

• $5^x = 125$ → $x = \log_5 125 = 3$ (since $5^3 = 125$)

Writing Expressions in Exponential Form and Solving for x

• Given: $\log_a x = b$

• Exponential Form: $x = a^b$

Examples:

• $\log_2 x = 3$ → $x = 2^3 = 8$

• $\log_2 x = 0.5$ → $x = 2^{0.5} = \sqrt{2}$

• $\log_5 x = 4$ → $x = 5^4 = 625$

Properties of Logarithms

Combining Logarithms

Logarithms can be combined or expanded using the following properties:

• Product Rule: $\log_a (xy) = \log_a x + \log_a y$

• Quotient Rule: $\log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y$

• Power Rule: $\log_a (x^r) = r \log_a x$

Example: $\log_7 5 + \log_7 2 - \log_7 10 = \log_7 \left(\frac{5 \cdot 2}{10}\right) = \log_7 1 = 0$

Expanding Logarithmic Expressions

To express logarithms as sums and differences of simpler logarithms, apply the properties above in reverse.

• Example: $\log_2 \left(\frac{x(x-1)^5}{x+1}\right)$

• Expand: $\log_2 x + 5\log_2(x-1) - \log_2(x+1)$

Solving Logarithmic and Exponential Equations

Solving Exponential Equations

• Isolate the exponential term if possible.

• Take the logarithm of both sides to solve for the variable.

Example: $2^x = 5$ → $x = \log_2 5$

Solving Logarithmic Equations

• Combine logarithms using properties if necessary.

• Rewrite the equation in exponential form to solve for the variable.

• Check for extraneous solutions (values that make the argument of any logarithm negative or zero are not valid).

Examples:

• $\log_3(x+4) = \log_3(2x-1)$ → $x+4 = 2x-1$ → $x = 5$

• $\log_2(x+4) + \log_2(x) = 1$ Combine: $\log_2[(x+4)x] = 1$ Exponential form: $(x+4)x = 2^1 = 2$ $x^2 + 4x - 2 = 0$ (solve quadratic equation)

Summary Table: Logarithm Properties

Property	Formula	Example
Product Rule	$\log_a(xy) = \log_a x + \log_a y$	$\log_2(8 \cdot 4) = \log_2 8 + \log_2 4$
Quotient Rule	$\log_a\left(\frac{x}{y}\right) = \log_a x - \log_a y$	$\log_5\left(\frac{25}{5}\right) = \log_5 25 - \log_5 5$
Power Rule	$\log_a(x^r) = r \log_a x$	$\log_3(9^2) = 2\log_3 9$

Additional info:

• Some expressions and equations were inferred from context due to partial or unclear text in the original file.

• All examples and explanations are standard for College Algebra and align with typical exam review content.