Logarithmic Properties and Expansions

This video tutorial covers the expansion of logarithmic expressions into sums or differences. It begins with a review of basic logarithmic properties, such as the power rule and product/quotient rules. The instructor then demonstrates how to expand complex logarithmic expressions, including those with exponents and radicals, through a series of examples. Each example illustrates the step-by-step process of breaking down the expression and applying the properties of logarithms to simplify it.

What is the expanded form of log(a^n)?

• n + log a

• log a^n

• log a + n

• n log a

How can log(a*b) be expanded?

• log a - log b

• log a + log b

• log a * log b

• log a / log b

In the expression log(xy/z), which term will have a negative sign?

• log x

• log xy

• log y

• log z

What is the expanded form of log(a^2b^3/c^4d^5)?

• log a + log b - log c - log d

• log a^2 + log b^3 - log c^4 - log d^5

• 2 log a + 3 log b + 4 log c + 5 log d

• 2 log a + 3 log b - 4 log c - 5 log d

Which of the following is a correct step in expanding log(a^2b^3/c^4d^5)?

• Add the exponents to the logs

• Move the exponents to the front

• Multiply the logs by the exponents

• Divide the logs by the exponents

How is the square root of a expressed as an exponent?

• a^2

• a^(1/3)

• a^1

• a^(1/2)

What is the expanded form of log(a^(1/2)b^(1/4)/c^(5/3))?

• log a + log b + log c

• log a^(1/2) + log b^(1/4) - log c^(5/3)

• log a + log b - log c

• 1/2 log a + 1/4 log b - 5/3 log c

What is the result of distributing the constant 3 in the expression 3(1/2 log a + 1/4 log b - 5/3 log c)?

• log a + log b - log c

• 3/2 log a + 3/4 log b - 5 log c

• 3 log a + 3 log b - 3 log c

• 1/2 log a + 1/4 log b - 5/3 log c

What happens to the expression 5/3 log c when multiplied by 3?

• It becomes 5 log c

• It becomes 3 log c

• It becomes 15 log c

• It becomes 5/3 log c

What is the final simplified form of the expression after distributing the constant?

• 3/2 log a + 3/4 log b - 5 log c

• 1/2 log a + 1/4 log b + 5/3 log c

• log a + log b - log c

• 3 log a + 3 log b + 3 log c