Evaluate 3 log base 2 of 4+1/2 log base 2 of 6-1/2 log base 2 of 24

$3 \log_{2} (4) + \frac{1}{2} \log_{2} (6) - \frac{1}{2} \log_{2} (24)$

Simplify each term.

Logarithm base $2$ of $4$ is $2$ .

$3 \cdot 2 + \frac{1}{2} \cdot \log_{2} (6) - \frac{1}{2} \cdot \log_{2} (24)$

Multiply $3$ by $2$ .

$6 + \frac{1}{2} \cdot \log_{2} (6) - \frac{1}{2} \cdot \log_{2} (24)$

Simplify $\frac{1}{2} \log_{2} (6)$ by moving $\frac{1}{2}$ inside the logarithm.

$6 + \log_{2} (6^{\frac{1}{2}}) - \frac{1}{2} \cdot \log_{2} (24)$

Multiply $- \frac{1}{2} \log_{2} (24)$ .

Reorder $\log_{2} (24)$ and $\frac{1}{2}$ .

$6 + \log_{2} (6^{\frac{1}{2}}) - (\frac{1}{2} \log_{2} (24))$

Simplify $\frac{1}{2} \log_{2} (24)$ by moving $\frac{1}{2}$ inside the logarithm.

$6 + \log_{2} (6^{\frac{1}{2}}) - \log_{2} (24^{\frac{1}{2}})$

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$6 + \log_{2} (\frac{6^{\frac{1}{2}}}{24^{\frac{1}{2}}})$

Rewrite $\log_{2} (\frac{6^{\frac{1}{2}}}{24^{\frac{1}{2}}})$ using the change of base formula.

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$6 + \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$6 + \frac{\log (\frac{6^{\frac{1}{2}}}{24^{\frac{1}{2}}})}{\log (2)}$

The result can be shown in multiple forms.

Exact Form:

$6 + \frac{\log (\frac{6^{\frac{1}{2}}}{24^{\frac{1}{2}}})}{\log (2)}$

Decimal Form:

$5$