Precalculus Examples

$\ln (\frac{10 e^{3 x}}{9 x})$

Step 1

Rewrite $\ln (\frac{10 e^{3 x}}{9 x})$ as $\ln (10 e^{3 x}) - \ln (9 x)$ .

$\ln (10 e^{3 x}) - \ln (9 x)$

Step 2

Rewrite $\ln (10 e^{3 x})$ as $\ln (10) + \ln (e^{3 x})$ .

$\ln (10) + \ln (e^{3 x}) - \ln (9 x)$

Step 3

Expand $\ln (e^{3 x})$ by moving $3 x$ outside the logarithm.

$\ln (10) + 3 x \ln (e) - \ln (9 x)$

Step 4

The natural logarithm of $e$ is $1$ .

$\ln (10) + 3 x \cdot 1 - \ln (9 x)$

Step 5

Multiply $3$ by $1$ .

$\ln (10) + 3 x - \ln (9 x)$

Step 6

Rewrite $\ln (9 x)$ as $\ln (9) + \ln (x)$ .

$\ln (10) + 3 x - (\ln (9) + \ln (x))$

Step 7

Simplify each term.

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Step 7.1

Rewrite $\ln (10)$ as $\ln (2 (5))$ .

$\ln (2 (5)) + 3 x - (\ln (9) + \ln (x))$

Step 7.2

Rewrite $\ln (2 (5))$ as $\ln (2) + \ln (5)$ .

$\ln (2) + \ln (5) + 3 x - (\ln (9) + \ln (x))$

Step 7.3

Simplify each term.

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Step 7.3.1

Rewrite $\ln (9)$ as $\ln (3^{2})$ .

$\ln (2) + \ln (5) + 3 x - (\ln (3^{2}) + \ln (x))$

Step 7.3.2

Expand $\ln (3^{2})$ by moving $2$ outside the logarithm.

$\ln (2) + \ln (5) + 3 x - (2 \ln (3) + \ln (x))$

Step 7.4

Apply the distributive property.

$\ln (2) + \ln (5) + 3 x - (2 \ln (3)) - \ln (x)$

Step 7.5

Multiply $2$ by $- 1$ .

$\ln (2) + \ln (5) + 3 x - 2 \ln (3) - \ln (x)$