Precalculus Examples

$\frac{6}{7} \cdot (\log_{6} (\sqrt{108}) - 1 \log_{6} (\frac{2}{\sqrt{3}}) + 3 \log_{6} (9) + \log_{6} (256))$

Step 1

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

$\frac{6}{7} \cdot (\log_{6} (\sqrt{108} \cdot 256) - 1 \log_{6} (\frac{2}{\sqrt{3}}) + 3 \log_{6} (9))$

Step 2

Simplify each term.

Tap for more steps...

Step 2.1

Rewrite $108$ as $6^{2} \cdot 3$ .

Tap for more steps...

Step 2.1.1

Factor $36$ out of $108$ .

$\frac{6}{7} \cdot (\log_{6} (\sqrt{36 (3)} \cdot 256) - 1 \log_{6} (\frac{2}{\sqrt{3}}) + 3 \log_{6} (9))$

Step 2.1.2

Rewrite $36$ as $6^{2}$ .

$\frac{6}{7} \cdot (\log_{6} (\sqrt{6^{2} \cdot 3} \cdot 256) - 1 \log_{6} (\frac{2}{\sqrt{3}}) + 3 \log_{6} (9))$

Step 2.2

Pull terms out from under the radical.

$\frac{6}{7} \cdot (\log_{6} (6 \sqrt{3} \cdot 256) - 1 \log_{6} (\frac{2}{\sqrt{3}}) + 3 \log_{6} (9))$

Step 2.3

Multiply $256$ by $6$ .

$\frac{6}{7} \cdot (\log_{6} (1536 \sqrt{3}) - 1 \log_{6} (\frac{2}{\sqrt{3}}) + 3 \log_{6} (9))$

Step 2.4

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{6}{7} \cdot (\log_{6} (1536 \sqrt{3}) - 1 \log_{6} (\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}) + 3 \log_{6} (9))$

Step 2.5

Combine and simplify the denominator.

Tap for more steps...

Step 2.5.1

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{6}{7} \cdot (\log_{6} (1536 \sqrt{3}) - 1 \log_{6} (\frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}) + 3 \log_{6} (9))$

Step 2.5.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{6}{7} \cdot (\log_{6} (1536 \sqrt{3}) - 1 \log_{6} (\frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}) + 3 \log_{6} (9))$

Step 2.5.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{6}{7} \cdot (\log_{6} (1536 \sqrt{3}) - 1 \log_{6} (\frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}) + 3 \log_{6} (9))$

Step 2.5.4

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

$\frac{6}{7} \cdot (\log_{6} (1536 \sqrt{3}) - 1 \log_{6} (\frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}) + 3 \log_{6} (9))$

Step 2.5.5

Add $1$ and $1$ .

$\frac{6}{7} \cdot (\log_{6} (1536 \sqrt{3}) - 1 \log_{6} (\frac{2 \sqrt{3}}{{\sqrt{3}}^{2}}) + 3 \log_{6} (9))$

Step 2.5.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Tap for more steps...

Step 2.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{6}{7} \cdot (\log_{6} (1536 \sqrt{3}) - 1 \log_{6} (\frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}) + 3 \log_{6} (9))$

Step 2.5.6.2

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

$\frac{6}{7} \cdot (\log_{6} (1536 \sqrt{3}) - 1 \log_{6} (\frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}) + 3 \log_{6} (9))$

Step 2.5.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{6}{7} \cdot (\log_{6} (1536 \sqrt{3}) - 1 \log_{6} (\frac{2 \sqrt{3}}{3^{\frac{2}{2}}}) + 3 \log_{6} (9))$

Step 2.5.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.5.6.4.1

Cancel the common factor.

$\frac{6}{7} \cdot (\log_{6} (1536 \sqrt{3}) - 1 \log_{6} (\frac{2 \sqrt{3}}{3^{\frac{2}{2}}}) + 3 \log_{6} (9))$

Step 2.5.6.4.2

Rewrite the expression.

$\frac{6}{7} \cdot (\log_{6} (1536 \sqrt{3}) - 1 \log_{6} (\frac{2 \sqrt{3}}{3^{1}}) + 3 \log_{6} (9))$

Step 2.5.6.5

Evaluate the exponent.

$\frac{6}{7} \cdot (\log_{6} (1536 \sqrt{3}) - 1 \log_{6} (\frac{2 \sqrt{3}}{3}) + 3 \log_{6} (9))$

Step 2.6

Rewrite $- 1 \log_{6} (\frac{2 \sqrt{3}}{3})$ as $- \log_{6} (\frac{2 \sqrt{3}}{3})$ .

$\frac{6}{7} \cdot (\log_{6} (1536 \sqrt{3}) - \log_{6} (\frac{2 \sqrt{3}}{3}) + 3 \log_{6} (9))$

Step 2.7

Simplify $3 \log_{6} (9)$ by moving $3$ inside the logarithm.

$\frac{6}{7} \cdot (\log_{6} (1536 \sqrt{3}) - \log_{6} (\frac{2 \sqrt{3}}{3}) + \log_{6} (9^{3}))$

Step 2.8

Raise $9$ to the power of $3$ .

$\frac{6}{7} \cdot (\log_{6} (1536 \sqrt{3}) - \log_{6} (\frac{2 \sqrt{3}}{3}) + \log_{6} (729))$

Step 3

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

$\frac{6}{7} \cdot (\log_{6} (\frac{1536 \sqrt{3}}{\frac{2 \sqrt{3}}{3}}) + \log_{6} (729))$

Step 4

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

$\frac{6}{7} \cdot \log_{6} (\frac{1536 \sqrt{3}}{\frac{2 \sqrt{3}}{3}} \cdot 729)$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\frac{6}{7} \cdot \log_{6} (1536 \sqrt{3} \frac{3}{2 \sqrt{3}} \cdot 729)$

Step 6

Cancel the common factor of $2 \sqrt{3}$ .

Tap for more steps...

Step 6.1

Factor $2 \sqrt{3}$ out of $1536 \sqrt{3}$ .

$\frac{6}{7} \cdot \log_{6} (2 \sqrt{3} (768) \frac{3}{2 \sqrt{3}} \cdot 729)$

Step 6.2

Cancel the common factor.

$\frac{6}{7} \cdot \log_{6} (2 \sqrt{3} \cdot 768 \frac{3}{2 \sqrt{3}} \cdot 729)$

Step 6.3

Rewrite the expression.

$\frac{6}{7} \cdot \log_{6} (768 \cdot 3 \cdot 729)$

Step 7

Simplify the expression.

Tap for more steps...

Step 7.1

Multiply $768$ by $3$ .

$\frac{6}{7} \cdot \log_{6} (2304 \cdot 729)$

Step 7.2

Multiply $2304$ by $729$ .

$\frac{6}{7} \cdot \log_{6} (1679616)$

Step 7.3

Logarithm base $6$ of $1679616$ is $8$ .

$\frac{6}{7} \cdot 8$

Step 8

Multiply $\frac{6}{7} \cdot 8$ .

Tap for more steps...

Step 8.1

Combine $\frac{6}{7}$ and $8$ .

$\frac{6 \cdot 8}{7}$

Step 8.2

Multiply $6$ by $8$ .

$\frac{48}{7}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{48}{7}$

Decimal Form:

$6.\overline{857142}$

Mixed Number Form:

$6 \frac{6}{7}$