Precalculus Examples

$3 \log ({(3)}^{2 x - 2}) + 4 \log ({(3)}^{4 x - 7})$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

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

$\log ({(3^{2 x - 2})}^{3}) + 4 \log ({(3)}^{4 x - 7})$

Step 1.2

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

Tap for more steps...

Step 1.2.1

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

$\log (3^{(2 x - 2) \cdot 3}) + 4 \log ({(3)}^{4 x - 7})$

Step 1.2.2

Apply the distributive property.

$\log (3^{2 x \cdot 3 - 2 \cdot 3}) + 4 \log ({(3)}^{4 x - 7})$

Step 1.2.3

Multiply $3$ by $2$ .

$\log (3^{6 x - 2 \cdot 3}) + 4 \log ({(3)}^{4 x - 7})$

Step 1.2.4

Multiply $- 2$ by $3$ .

$\log (3^{6 x - 6}) + 4 \log ({(3)}^{4 x - 7})$

Step 1.3

Simplify $4 \log (3^{4 x - 7})$ by moving $4$ inside the logarithm.

$\log (3^{6 x - 6}) + \log ({(3^{4 x - 7})}^{4})$

Step 1.4

Multiply the exponents in ${(3^{4 x - 7})}^{4}$ .

Tap for more steps...

Step 1.4.1

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

$\log (3^{6 x - 6}) + \log (3^{(4 x - 7) \cdot 4})$

Step 1.4.2

Apply the distributive property.

$\log (3^{6 x - 6}) + \log (3^{4 x \cdot 4 - 7 \cdot 4})$

Step 1.4.3

Multiply $4$ by $4$ .

$\log (3^{6 x - 6}) + \log (3^{16 x - 7 \cdot 4})$

Step 1.4.4

Multiply $- 7$ by $4$ .

$\log (3^{6 x - 6}) + \log (3^{16 x - 28})$

Step 2

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

$\log (3^{6 x - 6} \cdot 3^{16 x - 28})$

Step 3

Multiply $3^{6 x - 6}$ by $3^{16 x - 28}$ by adding the exponents.

Tap for more steps...

Step 3.1

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

$\log (3^{6 x - 6 + 16 x - 28})$

Step 3.2

Add $6 x$ and $16 x$ .

$\log (3^{22 x - 6 - 28})$

Step 3.3

Subtract $28$ from $- 6$ .

$\log (3^{22 x - 34})$