Logarithm Properties and Evaluations

The video tutorial explains how to evaluate a logarithmic expression using given values for natural logs of variables X, Y, and Z. It covers the application of logarithm properties, such as the power and quotient properties, to expand and simplify the expression. The tutorial emphasizes the importance of correctly applying these properties, especially when dealing with exponents. Finally, it demonstrates the substitution of given values to evaluate the expression, resulting in a final answer of negative 125.

What are the given values for natural log X, Y, and Z?

• ln X = 2, ln Y = 5, ln Z = 3

• ln X = 5, ln Y = 3, ln Z = 2

• ln X = 3, ln Y = 2, ln Z = 5

• ln X = 2, ln Y = 3, ln Z = 5

What is the value of ln Z in the given problem?

• 2

• 3

• 1

• 5

Which property of logarithms allows you to move an exponent in front of the log?

• Product Property

• Power Property

• Quotient Property

• Change of Base Property

How can the expression ln(Z^-1) be rewritten using the power property?

• ln Z + 1

• ln Z^-1

• 1/ln Z

• -1 * ln Z

What is the result of expanding ln(X/Y)^-1 using the quotient property?

• -ln X - ln Y

• ln X + ln Y

• -ln X + ln Y

• ln X - ln Y

What is the first step in expanding the expression ln(X/Y)^-1?

• Simplify to ln(XY)

• Directly substitute values

• Use the quotient property

• Apply the power property

Which property of logarithms is used to expand ln(XY)?

• Quotient Property

• Product Property

• Power Property

• Change of Base Property

What is the final expression after substituting the given values for ln X, ln Y, and ln Z?

• -25

• 25

• 125

• -125

What is the result of squaring the expression (ln X - ln Y)?

• ln(X - Y)^2

• (ln X - ln Y)^2

• ln X^2 - ln Y^2

• (ln X)^2 - (ln Y)^2

What is the final step in evaluating the expression?

• Substitute the values and calculate

• Apply the power property

• Expand using logarithm properties

• Simplify the expression