Write as a Single Logarithm 2/3* log base 2 of 27+5/2* log base 2 of 4

$\frac{2}{3} \cdot \log_{2} (27) + \frac{5}{2} \cdot \log_{2} (4)$

Simplify $\frac{2}{3} \log_{2} (27)$ by moving $\frac{2}{3}$ inside the logarithm.

$\log_{2} (27^{\frac{2}{3}}) + \frac{5}{2} \cdot \log_{2} (4)$

Rewrite $27$ as $3^{3}$ .

$\log_{2} ({(3^{3})}^{\frac{2}{3}}) + \frac{5}{2} \cdot \log_{2} (4)$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log_{2} (3^{3 (\frac{2}{3})}) + \frac{5}{2} \cdot \log_{2} (4)$

Cancel the common factor of $3$ .

Cancel the common factor.

$\log_{2} (3^{3 (\frac{2}{3})}) + \frac{5}{2} \cdot \log_{2} (4)$

Rewrite the expression.

$\log_{2} (3^{2}) + \frac{5}{2} \cdot \log_{2} (4)$

Raise $3$ to the power of $2$ .

$\log_{2} (9) + \frac{5}{2} \cdot \log_{2} (4)$

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

$\log_{2} (9) + \frac{5}{2} \cdot 2$

Cancel the common factor of $2$ .

Cancel the common factor.

$\log_{2} (9) + \frac{5}{2} \cdot 2$

Rewrite the expression.

$\log_{2} (9) + 5$

The result can be shown in multiple forms.

Exact Form:

$\log_{2} (9) + 5$

Decimal Form:

$8.16992500 \dots$