Finite Math Examples

$\log_{b} (x) = \frac{2}{3} \cdot \log_{b} (27) + 2 \log_{b} (2) - \log_{b} (3)$

Step 1

Simplify the right side.

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Step 1.1

Combine $\frac{2}{3}$ and $\log_{b} (27)$ .

$\log_{b} (x) = \frac{2 \log_{b} (27)}{3} + 2 \log_{b} (2) - \log_{b} (3)$

Step 2

Simplify the right side.

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Step 2.1

Simplify $\frac{2 \log_{b} (27)}{3} + 2 \log_{b} (2) - \log_{b} (3)$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Rewrite $\frac{2 \log_{b} (27)}{3}$ as $\frac{2}{3} \log_{b} (27)$ .

$\log_{b} (x) = \frac{2}{3} \log_{b} (27) + 2 \log_{b} (2) - \log_{b} (3)$

Step 2.1.1.2

Simplify $\frac{2}{3} \log_{b} (27)$ by moving $\frac{2}{3}$ inside the logarithm.

$\log_{b} (x) = \log_{b} (27^{\frac{2}{3}}) + 2 \log_{b} (2) - \log_{b} (3)$

Step 2.1.1.3

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

$\log_{b} (x) = \log_{b} ({(3^{3})}^{\frac{2}{3}}) + 2 \log_{b} (2) - \log_{b} (3)$

Step 2.1.1.4

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

$\log_{b} (x) = \log_{b} (3^{3 (\frac{2}{3})}) + 2 \log_{b} (2) - \log_{b} (3)$

Step 2.1.1.5

Cancel the common factor of $3$ .

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Step 2.1.1.5.1

Cancel the common factor.

$\log_{b} (x) = \log_{b} (3^{3 (\frac{2}{3})}) + 2 \log_{b} (2) - \log_{b} (3)$

Step 2.1.1.5.2

Rewrite the expression.

$\log_{b} (x) = \log_{b} (3^{2}) + 2 \log_{b} (2) - \log_{b} (3)$

Step 2.1.1.6

Raise $3$ to the power of $2$ .

$\log_{b} (x) = \log_{b} (9) + 2 \log_{b} (2) - \log_{b} (3)$

Step 2.1.1.7

Simplify $2 \log_{b} (2)$ by moving $2$ inside the logarithm.

$\log_{b} (x) = \log_{b} (9) + \log_{b} (2^{2}) - \log_{b} (3)$

Step 2.1.1.8

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

$\log_{b} (x) = \log_{b} (9) + \log_{b} (4) - \log_{b} (3)$

Step 2.1.2

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

$\log_{b} (x) = \log_{b} (9 \cdot 4) - \log_{b} (3)$

Step 2.1.3

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

$\log_{b} (x) = \log_{b} (\frac{9 \cdot 4}{3})$

Step 2.1.4

Cancel the common factor of $9$ and $3$ .

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Step 2.1.4.1

Factor $3$ out of $9 \cdot 4$ .

$\log_{b} (x) = \log_{b} (\frac{3 (3 \cdot 4)}{3})$

Step 2.1.4.2

Cancel the common factors.

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Step 2.1.4.2.1

Factor $3$ out of $3$ .

$\log_{b} (x) = \log_{b} (\frac{3 (3 \cdot 4)}{3 (1)})$

Step 2.1.4.2.2

Cancel the common factor.

$\log_{b} (x) = \log_{b} (\frac{3 (3 \cdot 4)}{3 \cdot 1})$

Step 2.1.4.2.3

Rewrite the expression.

$\log_{b} (x) = \log_{b} (\frac{3 \cdot 4}{1})$

Step 2.1.4.2.4

Divide $3 \cdot 4$ by $1$ .

$\log_{b} (x) = \log_{b} (3 \cdot 4)$

Step 2.1.5

Multiply $3$ by $4$ .

$\log_{b} (x) = \log_{b} (12)$

Step 3

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$x = 12$