Simplify/Condense log base 3 of 2x+3 log base 3 of 4x

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

Step 1

Simplify each term.

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

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

Apply the product rule to $4 x$ .

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

Raise $4$ to the power of $3$ .

$\log_{3} (2 x) + \log_{3} (64 x^{3})$

Step 2

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

$\log_{3} (2 x (64 x^{3}))$

Step 3

Rewrite using the commutative property of multiplication.

$\log_{3} (2 \cdot 64 x \cdot x^{3})$

Step 4

Multiply $x$ by $x^{3}$ by adding the exponents.

Move $x^{3}$ .

$\log_{3} (2 \cdot 64 (x^{3} x))$

Multiply $x^{3}$ by $x$ .

Raise $x$ to the power of $1$ .

$\log_{3} (2 \cdot 64 (x^{3} x^{1}))$

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

$\log_{3} (2 \cdot 64 x^{3 + 1})$

Add $3$ and $1$ .

$\log_{3} (2 \cdot 64 x^{4})$

Step 5

Multiply $2$ by $64$ .

$\log_{3} (128 x^{4})$