Questions: Use properties of logarithms to condense the logarithmic expression. Write the expression as 1. Where possible, evaluate logarithmic expressions. 1/2(log7 x + log7 y) 1/2(log7 x + log7 y) =

Use properties of logarithms to condense the logarithmic expression. Write the expression as 1. Where possible, evaluate logarithmic expressions.

1/2(log7 x + log7 y)

1/2(log7 x + log7 y) =

Transcript text: Use properties of logarithms to condense the logarithmic expression. Write the exp is 1 . Where possible, evaluate logarithmic expressions. \[ \frac{1}{2}\left(\log _{7} x+\log _{7} y\right) \] \[ \frac{1}{2}\left(\log _{7} x+\log _{7} y\right)= \]

Solution

Solution Steps

To condense the given logarithmic expression, we can use the properties of logarithms. Specifically, we will use the product rule for logarithms and the power rule for logarithms. The product rule states that \(\log_b(m) + \log_b(n) = \log_b(m \cdot n)\), and the power rule states that \(c \cdot \log_b(m) = \log_b(m^c)\).

Apply the product rule to combine \(\log_7(x) + \log_7(y)\) into a single logarithm.
Apply the power rule to bring the \(\frac{1}{2}\) inside the logarithm.

Step 1: Apply the Product Rule

Using the product rule for logarithms, we combine the logarithmic expressions: \[ \log_7(x) + \log_7(y) = \log_7(xy) \] Substituting the values \(x = 10\) and \(y = 5\): \[ \log_7(10 \cdot 5) = \log_7(50) \]

Step 2: Apply the Power Rule

Next, we apply the power rule to the expression: \[ \frac{1}{2} \log_7(50) = \log_7(50^{\frac{1}{2}}) = \log_7(\sqrt{50}) \]

Step 3: Evaluate the Logarithmic Expression

Calculating the value of \(\frac{1}{2} \log_7(50)\) gives us approximately: \[ \frac{1}{2} \log_7(50) \approx 1.0052 \]