Write as a Single Logarithm 2 log of 9-5 log of 3+ log of 4

$2 \log (9) - 5 \log (3) + \log (4)$

Step 1

Simplify each term.

Simplify $2 \log (9)$ by moving $2$ inside the logarithm.

$\log (9^{2}) - 5 \log (3) + \log (4)$

Raise $9$ to the power of $2$ .

$\log (81) - 5 \log (3) + \log (4)$

Simplify $- 5 \log (3)$ by moving $5$ inside the logarithm.

$\log (81) - \log (3^{5}) + \log (4)$

Raise $3$ to the power of $5$ .

$\log (81) - \log (243) + \log (4)$

Step 2

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

$\log (\frac{81}{243}) + \log (4)$

Step 3

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

$\log (\frac{81}{243} \cdot 4)$

Step 4

Cancel the common factor of $81$ and $243$ .

Factor $81$ out of $81$ .

$\log (\frac{81 (1)}{243} \cdot 4)$

Cancel the common factors.

Factor $81$ out of $243$ .

$\log (\frac{81 \cdot 1}{81 \cdot 3} \cdot 4)$

Cancel the common factor.

$\log (\frac{81 \cdot 1}{81 \cdot 3} \cdot 4)$

Rewrite the expression.

$\log (\frac{1}{3} \cdot 4)$

Step 5

Combine $\frac{1}{3}$ and $4$ .

$\log (\frac{4}{3})$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\log (\frac{4}{3})$

Decimal Form:

$0.12493873 \dots$