Precalculus Examples

$2 \log (\frac{\sqrt[4]{m}}{m}) + \log (\sqrt{100 m^{3}})$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Simplify $2 \log (\frac{\sqrt[4]{m}}{m})$ by moving $2$ inside the logarithm.

$\log ({(\frac{\sqrt[4]{m}}{m})}^{2}) + \log (\sqrt{100 m^{3}})$

Step 1.2

Apply the product rule to $\frac{\sqrt[4]{m}}{m}$ .

$\log (\frac{{\sqrt[4]{m}}^{2}}{m^{2}}) + \log (\sqrt{100 m^{3}})$

Step 1.3

Rewrite ${\sqrt[4]{m}}^{2}$ as $\sqrt{m}$ .

Tap for more steps...

Step 1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{m}$ as $m^{\frac{1}{4}}$ .

$\log (\frac{{(m^{\frac{1}{4}})}^{2}}{m^{2}}) + \log (\sqrt{100 m^{3}})$

Step 1.3.2

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

$\log (\frac{m^{\frac{1}{4} \cdot 2}}{m^{2}}) + \log (\sqrt{100 m^{3}})$

Step 1.3.3

Combine $\frac{1}{4}$ and $2$ .

$\log (\frac{m^{\frac{2}{4}}}{m^{2}}) + \log (\sqrt{100 m^{3}})$

Step 1.3.4

Cancel the common factor of $2$ and $4$ .

Tap for more steps...

Step 1.3.4.1

Factor $2$ out of $2$ .

$\log (\frac{m^{\frac{2 (1)}{4}}}{m^{2}}) + \log (\sqrt{100 m^{3}})$

Step 1.3.4.2

Cancel the common factors.

Tap for more steps...

Step 1.3.4.2.1

Factor $2$ out of $4$ .

$\log (\frac{m^{\frac{2 \cdot 1}{2 \cdot 2}}}{m^{2}}) + \log (\sqrt{100 m^{3}})$

Step 1.3.4.2.2

Cancel the common factor.

$\log (\frac{m^{\frac{2 \cdot 1}{2 \cdot 2}}}{m^{2}}) + \log (\sqrt{100 m^{3}})$

Step 1.3.4.2.3

Rewrite the expression.

$\log (\frac{m^{\frac{1}{2}}}{m^{2}}) + \log (\sqrt{100 m^{3}})$

Step 1.3.5

Rewrite $m^{\frac{1}{2}}$ as $\sqrt{m}$ .

$\log (\frac{\sqrt{m}}{m^{2}}) + \log (\sqrt{100 m^{3}})$

Step 1.4

Rewrite $100 m^{3}$ as ${(10 m)}^{2} m$ .

Tap for more steps...

Step 1.4.1

Rewrite $100$ as $10^{2}$ .

$\log (\frac{\sqrt{m}}{m^{2}}) + \log (\sqrt{10^{2} m^{3}})$

Step 1.4.2

Factor out $m^{2}$ .

$\log (\frac{\sqrt{m}}{m^{2}}) + \log (\sqrt{10^{2} (m^{2} m)})$

Step 1.4.3

Rewrite $10^{2} m^{2}$ as ${(10 m)}^{2}$ .

$\log (\frac{\sqrt{m}}{m^{2}}) + \log (\sqrt{{(10 m)}^{2} m})$

Step 1.5

Pull terms out from under the radical.

$\log (\frac{\sqrt{m}}{m^{2}}) + \log (10 m \sqrt{m})$

Step 2

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

$\log (\frac{\sqrt{m}}{m^{2}} (10 m \sqrt{m}))$

Step 3

Rewrite using the commutative property of multiplication.

$\log (10 \frac{\sqrt{m}}{m^{2}} (m \sqrt{m}))$

Step 4

Combine $10$ and $\frac{\sqrt{m}}{m^{2}}$ .

$\log (\frac{10 \sqrt{m}}{m^{2}} (m \sqrt{m}))$

Step 5

Cancel the common factor of $m$ .

Tap for more steps...

Step 5.1

Factor $m$ out of $m^{2}$ .

$\log (\frac{10 \sqrt{m}}{m \cdot m} (m \sqrt{m}))$

Step 5.2

Factor $m$ out of $m \sqrt{m}$ .

$\log (\frac{10 \sqrt{m}}{m \cdot m} (m (\sqrt{m})))$

Step 5.3

Cancel the common factor.

$\log (\frac{10 \sqrt{m}}{m \cdot m} (m \sqrt{m}))$

Step 5.4

Rewrite the expression.

$\log (\frac{10 \sqrt{m}}{m} \sqrt{m})$

Step 6

Combine $\frac{10 \sqrt{m}}{m}$ and $\sqrt{m}$ .

$\log (\frac{10 \sqrt{m} \sqrt{m}}{m})$

Step 7

Raise $\sqrt{m}$ to the power of $1$ .

$\log (\frac{10 ({\sqrt{m}}^{1} \sqrt{m})}{m})$

Step 8

Raise $\sqrt{m}$ to the power of $1$ .

$\log (\frac{10 ({\sqrt{m}}^{1} {\sqrt{m}}^{1})}{m})$

Step 9

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\log (\frac{10 {\sqrt{m}}^{1 + 1}}{m})$

Step 10

Add $1$ and $1$ .

$\log (\frac{10 {\sqrt{m}}^{2}}{m})$

Step 11

Rewrite ${\sqrt{m}}^{2}$ as $m$ .

Tap for more steps...

Step 11.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{m}$ as $m^{\frac{1}{2}}$ .

$\log (\frac{10 {(m^{\frac{1}{2}})}^{2}}{m})$

Step 11.2

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

$\log (\frac{10 m^{\frac{1}{2} \cdot 2}}{m})$

Step 11.3

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

$\log (\frac{10 m^{\frac{2}{2}}}{m})$

Step 11.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 11.4.1

Cancel the common factor.

$\log (\frac{10 m^{\frac{2}{2}}}{m})$

Step 11.4.2

Rewrite the expression.

$\log (\frac{10 m^{1}}{m})$

Step 11.5

Simplify.

$\log (\frac{10 m}{m})$

Step 12

Cancel the common factor of $m$ .

Tap for more steps...

Step 12.1

Cancel the common factor.

$\log (\frac{10 m}{m})$

Step 12.2

Divide $10$ by $1$ .

$\log (10)$

Step 13

Logarithm base $10$ of $10$ is $1$ .

$1$