Algebra Examples

$\frac{1}{2} \cdot \log_{3} (9 x) + \frac{3}{2} \cdot \log_{3} (4 x^{3})$

Step 1

Simplify each term.

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Step 1.1

Simplify $\frac{1}{2} \log_{3} (9 x)$ by moving $\frac{1}{2}$ inside the logarithm.

$\log_{3} ({(9 x)}^{\frac{1}{2}}) + \frac{3}{2} \cdot \log_{3} (4 x^{3})$

Step 1.2

Apply the product rule to $9 x$ .

$\log_{3} (9^{\frac{1}{2}} x^{\frac{1}{2}}) + \frac{3}{2} \cdot \log_{3} (4 x^{3})$

Step 1.3

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

$\log_{3} ({(3^{2})}^{\frac{1}{2}} x^{\frac{1}{2}}) + \frac{3}{2} \cdot \log_{3} (4 x^{3})$

Step 1.4

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

$\log_{3} (3^{2 (\frac{1}{2})} x^{\frac{1}{2}}) + \frac{3}{2} \cdot \log_{3} (4 x^{3})$

Step 1.5

Cancel the common factor of $2$ .

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Step 1.5.1

Cancel the common factor.

$\log_{3} (3^{2 (\frac{1}{2})} x^{\frac{1}{2}}) + \frac{3}{2} \cdot \log_{3} (4 x^{3})$

Step 1.5.2

Rewrite the expression.

$\log_{3} (3^{1} x^{\frac{1}{2}}) + \frac{3}{2} \cdot \log_{3} (4 x^{3})$

Step 1.6

Evaluate the exponent.

$\log_{3} (3 x^{\frac{1}{2}}) + \frac{3}{2} \cdot \log_{3} (4 x^{3})$

Step 1.7

Simplify $\frac{3}{2} \log_{3} (4 x^{3})$ by moving $\frac{3}{2}$ inside the logarithm.

$\log_{3} (3 x^{\frac{1}{2}}) + \log_{3} ({(4 x^{3})}^{\frac{3}{2}})$

Step 1.8

Apply the product rule to $4 x^{3}$ .

$\log_{3} (3 x^{\frac{1}{2}}) + \log_{3} (4^{\frac{3}{2}} {(x^{3})}^{\frac{3}{2}})$

Step 1.9

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

$\log_{3} (3 x^{\frac{1}{2}}) + \log_{3} ({(2^{2})}^{\frac{3}{2}} {(x^{3})}^{\frac{3}{2}})$

Step 1.10

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

$\log_{3} (3 x^{\frac{1}{2}}) + \log_{3} (2^{2 (\frac{3}{2})} {(x^{3})}^{\frac{3}{2}})$

Step 1.11

Cancel the common factor of $2$ .

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Step 1.11.1

Cancel the common factor.

$\log_{3} (3 x^{\frac{1}{2}}) + \log_{3} (2^{2 (\frac{3}{2})} {(x^{3})}^{\frac{3}{2}})$

Step 1.11.2

Rewrite the expression.

$\log_{3} (3 x^{\frac{1}{2}}) + \log_{3} (2^{3} {(x^{3})}^{\frac{3}{2}})$

Step 1.12

Raise $2$ to the power of $3$ .

$\log_{3} (3 x^{\frac{1}{2}}) + \log_{3} (8 {(x^{3})}^{\frac{3}{2}})$

Step 1.13

Multiply the exponents in ${(x^{3})}^{\frac{3}{2}}$ .

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Step 1.13.1

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

$\log_{3} (3 x^{\frac{1}{2}}) + \log_{3} (8 x^{3 (\frac{3}{2})})$

Step 1.13.2

Multiply $3 (\frac{3}{2})$ .

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Step 1.13.2.1

Combine $3$ and $\frac{3}{2}$ .

$\log_{3} (3 x^{\frac{1}{2}}) + \log_{3} (8 x^{\frac{3 \cdot 3}{2}})$

Step 1.13.2.2

Multiply $3$ by $3$ .

$\log_{3} (3 x^{\frac{1}{2}}) + \log_{3} (8 x^{\frac{9}{2}})$

Step 2

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

$\log_{3} (3 x^{\frac{1}{2}} (8 x^{\frac{9}{2}}))$

Step 3

Rewrite using the commutative property of multiplication.

$\log_{3} (3 \cdot 8 x^{\frac{1}{2}} x^{\frac{9}{2}})$

Step 4

Multiply $x^{\frac{1}{2}}$ by $x^{\frac{9}{2}}$ by adding the exponents.

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Step 4.1

Move $x^{\frac{9}{2}}$ .

$\log_{3} (3 \cdot 8 (x^{\frac{9}{2}} x^{\frac{1}{2}}))$

Step 4.2

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

$\log_{3} (3 \cdot 8 x^{\frac{9}{2} + \frac{1}{2}})$

Step 4.3

Combine the numerators over the common denominator.

$\log_{3} (3 \cdot 8 x^{\frac{9 + 1}{2}})$

Step 4.4

Add $9$ and $1$ .

$\log_{3} (3 \cdot 8 x^{\frac{10}{2}})$

Step 4.5

Divide $10$ by $2$ .

$\log_{3} (3 \cdot 8 x^{5})$

Step 5

Multiply $3$ by $8$ .

$\log_{3} (24 x^{5})$