Questions: Write the expression as a single logarithm. 4 loga(8z+1)+1/2 loga(z+7) loga((8z+1)^4(z+7)^1/2)

Transcript text: Write the expression as a single logarithm. \[ 4 \log _{a}(8 z+1)+\frac{1}{2} \log _{a}(z+7) \] \[ \log _{a}\left((8 z+1)^{4}(z+7)^{\frac{1}{2}}\right) \]

Solution

Solution Steps

Step 1: Apply the Power Rule of Logarithms

The power rule of logarithms states that \( k \log_a(b) = \log_a(b^k) \). Apply this rule to both terms in the expression: \[ 4 \log_a(8z + 1) = \log_a((8z + 1)^4) \] \[ \frac{1}{2} \log_a(z + 7) = \log_a((z + 7)^{\frac{1}{2}}) \]

Step 2: Combine the Logarithms Using the Product Rule

The product rule of logarithms states that \( \log_a(b) + \log_a(c) = \log_a(bc) \). Combine the two logarithms: \[ \log_a((8z + 1)^4) + \log_a((z + 7)^{\frac{1}{2}}) = \log_a\left((8z + 1)^4 \cdot (z + 7)^{\frac{1}{2}}\right) \]