Precalculus Examples

$- 3 \ln (5 x^{3}) + 4 \ln (3 x^{4})$

Step 1

Simplify $- 3 \ln (5 x^{3})$ by moving $3$ inside the logarithm.

$- \ln ({(5 x^{3})}^{3}) + 4 \ln (3 x^{4})$

Step 2

Apply the product rule to $5 x^{3}$ .

$- \ln (5^{3} {(x^{3})}^{3}) + 4 \ln (3 x^{4})$

Step 3

Raise $5$ to the power of $3$ .

$- \ln (125 {(x^{3})}^{3}) + 4 \ln (3 x^{4})$

Step 4

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

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Step 4.1

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

$- \ln (125 x^{3 \cdot 3}) + 4 \ln (3 x^{4})$

Step 4.2

Multiply $3$ by $3$ .

$- \ln (125 x^{9}) + 4 \ln (3 x^{4})$

Step 5

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

$- \ln (125 x^{9}) + \ln ({(3 x^{4})}^{4})$

Step 6

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

$- \ln (125 x^{9}) + \ln (3^{4} {(x^{4})}^{4})$

Step 7

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

$- \ln (125 x^{9}) + \ln (81 {(x^{4})}^{4})$

Step 8

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

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Step 8.1

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

$- \ln (125 x^{9}) + \ln (81 x^{4 \cdot 4})$

Step 8.2

Multiply $4$ by $4$ .

$- \ln (125 x^{9}) + \ln (81 x^{16})$