Write as a Single Logarithm 2* log of 6-1/4* log of 16+ log of 3

$2 \cdot \log (6) - \frac{1}{4} \cdot \log (16) + \log (3)$

Step 1

Simplify each term.

Simplify $2 \log (6)$ by moving $2$ inside the logarithm.

$\log (6^{2}) - \frac{1}{4} \cdot \log (16) + \log (3)$

Raise $6$ to the power of $2$ .

$\log (36) - \frac{1}{4} \cdot \log (16) + \log (3)$

Multiply $- \frac{1}{4} \log (16)$ .

Reorder $\log (16)$ and $\frac{1}{4}$ .

$\log (36) - (\frac{1}{4} \log (16)) + \log (3)$

Simplify $\frac{1}{4} \log (16)$ by moving $\frac{1}{4}$ inside the logarithm.

$\log (36) - \log (16^{\frac{1}{4}}) + \log (3)$

Rewrite $16$ as $2^{4}$ .

$\log (36) - \log ({(2^{4})}^{\frac{1}{4}}) + \log (3)$

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

$\log (36) - \log (2^{4 (\frac{1}{4})}) + \log (3)$

Cancel the common factor of $4$ .

Cancel the common factor.

$\log (36) - \log (2^{4 (\frac{1}{4})}) + \log (3)$

Rewrite the expression.

$\log (36) - \log (2^{1}) + \log (3)$

Evaluate the exponent.

$\log (36) - \log (2) + \log (3)$

Step 2

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

$\log (\frac{36}{2}) + \log (3)$

Step 3

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log (\frac{36}{2} \cdot 3)$

Step 4

Divide $36$ by $2$ .

$\log (18 \cdot 3)$

Step 5

Multiply $18$ by $3$ .

$\log (54)$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\log (54)$

Decimal Form:

$1.73239375 \dots$