Questions: Use the properties of logs to condense the expression: 4[ln z+ln (z+5)]-2 ln (z+5)=

Transcript text: Use the properties of logs to condense the expression: \[ 4[\ln z+\ln (z+5)]-2 \ln (z+5)= \]

Solution

Solution Steps

To condense the given logarithmic expression, we can use the properties of logarithms. First, apply the distributive property to the terms inside the brackets. Then, use the product rule of logarithms to combine the terms inside the brackets. Finally, apply the power rule and the quotient rule to simplify the expression further.

Step 1: Expand the Expression

We start with the expression: \[ 4[\ln z + \ln(z + 5)] - 2 \ln(z + 5) \] Using the distributive property, we can expand this to: \[ 4 \ln z + 4 \ln(z + 5) - 2 \ln(z + 5) \]

Step 2: Combine Like Terms

Next, we combine the logarithmic terms involving \(\ln(z + 5)\): \[ 4 \ln z + (4 - 2) \ln(z + 5) = 4 \ln z + 2 \ln(z + 5) \]

Step 3: Final Condensed Form

The expression is now simplified to: \[ 4 \ln z + 2 \ln(z + 5) \] This is the condensed form of the original expression.