GIVEN:
The following values are given:
[tex]\begin{gathered} \log _2x=a \\ \log _2y=b \end{gathered}[/tex]
We are to evaluate:
[tex]\log _2x^2y[/tex]
CALCULATION:
Step 1: Apply the law of logarithm
[tex]\log _z(m\times n)=\log _zm+\log _zn[/tex]
Therefore, we have
[tex]\log _2x^2y=\log _2x^2+\log _2y[/tex]
Step 2: Apply the law of logarithm
[tex]\log _am^n=n\log _am[/tex]
Therefore, the first expression becomes:
[tex]\log _2x^2=2\log _2x[/tex]
Hence, the expression becomes:
[tex]\Rightarrow2\log _2x+\log _2y[/tex]
Step 3: Substitute for a and b in the expression above
[tex]\begin{gathered} \log _2x=a \\ \log _2y=b \end{gathered}[/tex]
Therefore, the expression becomes:
[tex]2\log _2x+\log _2y=2a+b[/tex]
ANSWER:
[tex]\log _2x^2y=2a+b[/tex]
