Algebra Examples

$\ln (7) - \frac{1}{3} \ln (27) + \ln (6)$

Step 1

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

$- \frac{1}{3} \cdot \ln (27) + \ln (7 \cdot 6)$

Step 2

Simplify each term.

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Step 2.1

Multiply $- \frac{1}{3} \ln (27)$ .

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Step 2.1.1

Reorder $\ln (27)$ and $\frac{1}{3}$ .

$- (\frac{1}{3} \ln (27)) + \ln (7 \cdot 6)$

Step 2.1.2

Simplify $\frac{1}{3} \ln (27)$ by moving $\frac{1}{3}$ inside the logarithm.

$- \ln (27^{\frac{1}{3}}) + \ln (7 \cdot 6)$

Step 2.2

Rewrite $27$ as $3^{3}$ .

$- \ln ({(3^{3})}^{\frac{1}{3}}) + \ln (7 \cdot 6)$

Step 2.3

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

$- \ln (3^{3 (\frac{1}{3})}) + \ln (7 \cdot 6)$

Step 2.4

Cancel the common factor of $3$ .

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Step 2.4.1

Cancel the common factor.

$- \ln (3^{3 (\frac{1}{3})}) + \ln (7 \cdot 6)$

Step 2.4.2

Rewrite the expression.

$- \ln (3^{1}) + \ln (7 \cdot 6)$

Step 2.5

Evaluate the exponent.

$- \ln (3) + \ln (7 \cdot 6)$

Step 2.6

Multiply $7$ by $6$ .

$- \ln (3) + \ln (42)$

Step 3

The result can be shown in multiple forms.

Exact Form:

$- \ln (3) + \ln (42)$

Decimal Form:

$2.63905732 \dots$