Simplify/Condense 1/3* log of 3x+2/3* log of 3x

$\frac{1}{3} \cdot \log (3 x) + \frac{2}{3} \cdot \log (3 x)$

Step 1

Simplify each term.

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

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

Apply the product rule to $3 x$ .

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

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

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

Apply the product rule to $3 x$ .

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

Step 2

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

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

Step 3

Rewrite using the commutative property of multiplication.

$\log (3^{\frac{1}{3}} \cdot 3^{\frac{2}{3}} x^{\frac{1}{3}} x^{\frac{2}{3}})$

Step 4

Multiply $3^{\frac{1}{3}}$ by $3^{\frac{2}{3}}$ by adding the exponents.

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

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

Combine the numerators over the common denominator.

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

Add $1$ and $2$ .

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

Divide $3$ by $3$ .

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

Step 5

Simplify $3^{1} x^{\frac{1}{3}} x^{\frac{2}{3}}$ .

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

Step 6

Multiply $x^{\frac{1}{3}}$ by $x^{\frac{2}{3}}$ by adding the exponents.

Move $x^{\frac{2}{3}}$ .

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

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

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

Combine the numerators over the common denominator.

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

Add $2$ and $1$ .

$\log (3 x^{\frac{3}{3}})$

Divide $3$ by $3$ .

$\log (3 x^{1})$

Step 7

Simplify $3 x^{1}$ .

$\log (3 x)$