Algebra Examples

$\log (a) + 2 \log (b) - \frac{1}{2} \log (9 c^{2})$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

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

$\log (a) + \log (b^{2}) - \frac{1}{2} \cdot \log (9 c^{2})$

Step 1.2

Multiply $- \frac{1}{2} \log (9 c^{2})$ .

Tap for more steps...

Step 1.2.1

Reorder $\log (9 c^{2})$ and $\frac{1}{2}$ .

$\log (a) + \log (b^{2}) - (\frac{1}{2} \log (9 c^{2}))$

Step 1.2.2

Simplify $\frac{1}{2} \log (9 c^{2})$ by moving $\frac{1}{2}$ inside the logarithm.

$\log (a) + \log (b^{2}) - \log ({(9 c^{2})}^{\frac{1}{2}})$

Step 1.3

Apply the product rule to $9 c^{2}$ .

$\log (a) + \log (b^{2}) - \log (9^{\frac{1}{2}} {(c^{2})}^{\frac{1}{2}})$

Step 1.4

Rewrite $9$ as $3^{2}$ .

$\log (a) + \log (b^{2}) - \log ({(3^{2})}^{\frac{1}{2}} {(c^{2})}^{\frac{1}{2}})$

Step 1.5

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

$\log (a) + \log (b^{2}) - \log (3^{2 (\frac{1}{2})} {(c^{2})}^{\frac{1}{2}})$

Step 1.6

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.6.1

Cancel the common factor.

$\log (a) + \log (b^{2}) - \log (3^{2 (\frac{1}{2})} {(c^{2})}^{\frac{1}{2}})$

Step 1.6.2

Rewrite the expression.

$\log (a) + \log (b^{2}) - \log (3^{1} {(c^{2})}^{\frac{1}{2}})$

Step 1.7

Evaluate the exponent.

$\log (a) + \log (b^{2}) - \log (3 {(c^{2})}^{\frac{1}{2}})$

Step 1.8

Multiply the exponents in ${(c^{2})}^{\frac{1}{2}}$ .

Tap for more steps...

Step 1.8.1

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

$\log (a) + \log (b^{2}) - \log (3 c^{2 (\frac{1}{2})})$

Step 1.8.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.8.2.1

Cancel the common factor.

$\log (a) + \log (b^{2}) - \log (3 c^{2 (\frac{1}{2})})$

Step 1.8.2.2

Rewrite the expression.

$\log (a) + \log (b^{2}) - \log (3 c^{1})$

Step 1.9

Simplify.

$\log (a) + \log (b^{2}) - \log (3 c)$

Step 2

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

$\log (a b^{2}) - \log (3 c)$

Step 3

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

$\log (\frac{a b^{2}}{3 c})$