Rewrite the following expression as a single logarithm.

Question

2 ln left parenthesis 2 a right parenthesis minus ln b
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Part 1
2 ln left parenthesis 2 a right parenthesis minus ln bequals
  
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Answer 1

To combine the expression \(2 \ln(2a) - \ln(b)\) into a single logarithm, we can use the properties of logarithms.

1. The property \(k \ln(x) = \ln(x^k)\) allows us to rewrite \(2 \ln(2a)\) as \(\ln((2a)^2)\).
2. The property \(\ln(x) - \ln(y) = \ln\left(\frac{x}{y}\right)\) allows us to combine the logarithms.

Let's apply these steps:

1. Rewrite \(2 \ln(2a)\): \[ 2 \ln(2a) = \ln((2a)^2) = \ln(4a^2) \]

2. Combine it with \(- \ln(b)\): \[ \ln(4a^2) - \ln(b) = \ln\left(\frac{4a^2}{b}\right) \]

So, the expression \(2 \ln(2a) - \ln(b)\) as a single logarithm is:

\[ \ln\left(\frac{4a^2}{b}\right) \]

This is your final result!