Question

given that:
`log_(2) \: x =a \: and \: log_(2) \: y= b\:` therefore ,express
`log_(2) {} \: x {}^(2) y \: in \: terms \: of \: a \: and \: b`

Answer 1

GIVEN:

The following values are given:

`\begin{gathered} \log _2x=a \\ \log _2y=b \end{gathered}`

We are to evaluate:

`\log _2x^2y`

CALCULATION:

Step 1: Apply the law of logarithm

`\log _z(m* n)=\log _zm+\log _zn`

Therefore, we have

`\log _2x^2y=\log _2x^2+\log _2y`

Step 2: Apply the law of logarithm

`\log _am^n=n\log _am`

Therefore, the first expression becomes:

`\log _2x^2=2\log _2x`

Hence, the expression becomes:

`\Rightarrow2\log _2x+\log _2y`

Step 3: Substitute for a and b in the expression above

`\begin{gathered} \log _2x=a \\ \log _2y=b \end{gathered}`

Therefore, the expression becomes:

`2\log _2x+\log _2y=2a+b`

ANSWER:

`\log _2x^2y=2a+b`