Precalculus Examples

$\ln (3) - 2 (\ln (4) + \ln (32))$

Step 1

Simplify each term.

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Step 1.1

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

$\ln (3) - 2 \ln (4 \cdot 32)$

Step 1.2

Multiply $4$ by $32$ .

$\ln (3) - 2 \ln (128)$

Step 1.3

Simplify $- 2 \ln (128)$ by moving $2$ inside the logarithm.

$\ln (3) - \ln (128^{2})$

Step 1.4

Raise $128$ to the power of $2$ .

$\ln (3) - \ln (16384)$

Step 2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{3}{16384})$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\ln (\frac{3}{16384})$

Decimal Form:

$- 8.60544823 \dots$