Math Problem Statement

Use properties of logarithms to expand each logarithmic expression as much as possible. Evaluate logarithmic expressions without using a calculator if possible. log Subscript b Baseline left parenthesis z cubed y right parenthesis Question content area bottom Part 1 log Subscript b Baseline left parenthesis z cubed y right parenthesis equals 3. log Subscript b Baseline left parenthesis z right parenthesis plus log Subscript b Baseline left parenthesis y right parenthesis

Solution

To expand logb(z3y)\log_b(z^3y) using the properties of logarithms, follow these steps:

Step 1: Apply the product rule of logarithms

The product rule states: logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y) Applying this rule: logb(z3y)=logb(z3)+logb(y)\log_b(z^3y) = \log_b(z^3) + \log_b(y)

Step 2: Apply the power rule of logarithms

The power rule states: logb(xn)=nlogb(x)\log_b(x^n) = n \cdot \log_b(x) Using this rule on logb(z3)\log_b(z^3): logb(z3)=3logb(z)\log_b(z^3) = 3 \cdot \log_b(z)

Step 3: Combine the terms

Substitute 3logb(z)3 \cdot \log_b(z) back into the expression: logb(z3y)=3logb(z)+logb(y)\log_b(z^3y) = 3 \cdot \log_b(z) + \log_b(y)

Final Answer:

logb(z3y)=3logb(z)+logb(y)\log_b(z^3y) = 3 \cdot \log_b(z) + \log_b(y)

If there are any specific values for logb(z)\log_b(z) or logb(y)\log_b(y), you can substitute them in to evaluate further.


Would you like additional examples or clarifications?

Related Questions:

  1. How do the product and power rules of logarithms differ?
  2. Can you simplify logb(xnym)\log_b(x^n \cdot y^m) using logarithmic rules?
  3. What happens if one of the terms is zero or undefined in a logarithmic expression?
  4. How do these rules apply to natural logarithms (ln\ln)?
  5. How do you solve logb(x)=c\log_b(x) = c for xx if bb and cc are known?

Tip:

Always check the domain of the variables in logarithmic expressions. For example, z>0z > 0 and y>0y > 0 are required for logb(z3y)\log_b(z^3y) to be defined.

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Math Problem Analysis

Mathematical Concepts

Logarithms
Properties of Logarithms
Algebra

Formulas

Product rule: log_b(xy) = log_b(x) + log_b(y)
Power rule: log_b(x^n) = n * log_b(x)

Theorems

Properties of Logarithms

Suitable Grade Level

Grades 9-11