Write as a Single Logarithm log base 9 of 10- log base 9 of 1/2- log base 9 of 4

$\log_{9} (10) - \log_{9} (\frac{1}{2}) - \log_{9} (4)$

Step 1

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

$\log_{9} (\frac{10}{\frac{1}{2}}) - \log_{9} (4)$

Step 2

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

$\log_{9} (\frac{\frac{10}{\frac{1}{2}}}{4})$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\log_{9} (\frac{10}{\frac{1}{2}} \cdot \frac{1}{4})$

Step 4

Cancel the common factor of $2$ .

Factor $2$ out of $10$ .

$\log_{9} (\frac{2 (5)}{\frac{1}{2}} \cdot \frac{1}{4})$

Factor $2$ out of $4$ .

$\log_{9} (\frac{2 \cdot 5}{\frac{1}{2}} \cdot \frac{1}{2 \cdot 2})$

Cancel the common factor.

$\log_{9} (\frac{2 \cdot 5}{\frac{1}{2}} \cdot \frac{1}{2 \cdot 2})$

Rewrite the expression.

$\log_{9} (\frac{5}{\frac{1}{2}} \cdot \frac{1}{2})$

Step 5

Multiply $\frac{5}{\frac{1}{2}}$ by $\frac{1}{2}$ .

$\log_{9} (\frac{5}{\frac{1}{2} \cdot 2})$

Step 6

Combine $\frac{1}{2}$ and $2$ .

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

Step 7

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$\log_{9} (\frac{5}{1})$

Step 8

Divide $5$ by $1$ .

$\log_{9} (5)$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\log_{9} (5)$

Decimal Form:

$0.73248676 \dots$