Math Problem Statement

Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where​ possible, evaluate logarithmic expressions. one half left parenthesis log Subscript 6 Baseline a plus log Subscript 6 Baseline b right parenthesis Question content area bottom Part 1 one half left parenthesis log Subscript 6 Baseline a plus log Subscript 6 Baseline b right parenthesis

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Solution

To condense the given logarithmic expression, follow these steps:

Expression:

12(log6a+log6b)\frac{1}{2} \left( \log_6 a + \log_6 b \right)

Step 1: Apply the Product Rule of logarithms

The product rule states: logbx+logby=logb(xy)\log_b x + \log_b y = \log_b (x \cdot y) So, log6a+log6b=log6(ab)\log_6 a + \log_6 b = \log_6 (a \cdot b)

Now the expression becomes: 12log6(ab)\frac{1}{2} \log_6 (a \cdot b)

Step 2: Apply the Power Rule of logarithms

The power rule states: clogbx=logb(xc)c \cdot \log_b x = \log_b (x^c) Here, c=12c = \frac{1}{2}. So: 12log6(ab)=log6((ab)12)\frac{1}{2} \log_6 (a \cdot b) = \log_6 \left( (a \cdot b)^{\frac{1}{2}} \right)

Step 3: Simplify

The exponent 12\frac{1}{2} means the square root. Thus: log6((ab)12)=log6ab\log_6 \left( (a \cdot b)^{\frac{1}{2}} \right) = \log_6 \sqrt{a \cdot b}

Final Answer:

log6ab\log_6 \sqrt{a \cdot b}


Would you like me to explain this in more detail or assist with a related topic? Here are 5 related questions:

  1. How do the properties of logarithms help simplify expressions in general?
  2. Can you provide examples where the base of the logarithm is not 6?
  3. How do you evaluate logarithms with specific values, like log636\log_6 36?
  4. What is the difference between condensing and expanding logarithmic expressions?
  5. How would this process differ if you had subtraction instead of addition inside the logarithm?

Tip: Always check for opportunities to apply logarithmic properties like the product, quotient, and power rules when simplifying expressions!

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Math Problem Analysis

Mathematical Concepts

Logarithms
Properties of Logarithms
Condensing Logarithmic Expressions

Formulas

log_b x + log_b y = log_b (x * y) (Product Rule)
c * log_b x = log_b (x^c) (Power Rule)

Theorems

Logarithm Product Rule
Logarithm Power Rule

Suitable Grade Level

Grades 10-12