Solve \((\log_{2} m)^{2} - \log_{2} m^{3} = 10\).
Question 1 Report
Solve \((\log_{2} m)^{2} - \log_{2} m^{3} = 10\).
Answer Details
Let \(u=\log_{2}m\). Using the power law of logarithms, \(\log_{2}m^{3}=3\log_{2}m=3u\). The equation becomes a quadratic in \(u\):
\[u^{2}-3u=10\;\Rightarrow\; u^{2}-3u-10=0.\]
Factorising,
\[(u-5)(u+2)=0\;\Rightarrow\; u=5\quad\text{or}\quad u=-2.\]
Now convert back to \(m\) using \(u=\log_{2}m\Rightarrow m=2^{u}\):
- \(u=5:\quad m=2^{5}=32.\)
- \(u=-2:\quad m=2^{-2}=\dfrac{1}{4}.\)
Both values are positive, so both are valid solutions:
\[m=32\quad\text{or}\quad m=\tfrac{1}{4}.\]
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