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Answer
The simplified form is \(4\log_{2}\left(\frac{a}{b^{3}}\right)\).
Solution
Calculate or simplify the expression \( 4 \log _{2} a-12 \log _{2} b \).
Simplify the expression by following steps:
- step0: Solution:
\(4\log_{2}{\left(a\right)}-12\log_{2}{\left(b\right)}\)
- step1: Use the logarithm base change rule:
\(\log_{2}{\left(a^{4}\right)}-\log_{2}{\left(b^{12}\right)}\)
- step2: Transform the expression:
\(\log_{2}{\left(\frac{a^{4}}{b^{12}}\right)}\)
- step3: Rearrange the terms:
\(\log_{2}{\left(\left(\frac{a}{b^{3}}\right)^{4}\right)}\)
- step4: Simplify the expression:
\(4\log_{2}{\left(\frac{a}{b^{3}}\right)}\)
The simplified form of the expression \(4 \log _{2} a-12 \log _{2} b\) is \(4\log_{2}\left(\frac{a}{b^{3}}\right)\).
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Simplify this solution Extra Insights
To simplify the expression \( 4 \log _{2} a - 12 \log _{2} b \), you can factor out the common logarithmic bases. So, let's rewrite it: \[ 4 \log_{2} a - 12 \log_{2} b = 4 (\log_{2} a - 3 \log_{2} b) \] Now, using the properties of logarithms, specifically that \( n \log_{b} x = \log_{b} (x^n) \) and that the difference of logs can be written as a single log of a quotient: \[ \log_{2} a - 3 \log_{2} b = \log_{2} a - \log_{2} (b^3) = \log_{2} \left( \frac{a}{b^3} \right) \] So, putting it all together, we can express the whole expression as: \[ 4 \log_{2} \left( \frac{a}{b^3} \right) \] This is the simplified form of the original expression!
