Finite Math Examples

$\log_{b} (x) = \frac{3}{2} \cdot \log_{b} (9) - \frac{2}{3} \cdot \log_{b} (27)$

Step 1

Simplify the right side.

Tap for more steps...

Step 1.1

Simplify each term.

Tap for more steps...

Step 1.1.1

Combine $\frac{3}{2}$ and $\log_{b} (9)$ .

$\log_{b} (x) = \frac{3 \log_{b} (9)}{2} - \frac{2}{3} \log_{b} (27)$

Step 1.1.2

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

$\log_{b} (x) = \frac{3 \log_{b} (9)}{2} - \frac{\log_{b} (27) \cdot 2}{3}$

Step 1.1.3

Move $2$ to the left of $\log_{b} (27)$ .

$\log_{b} (x) = \frac{3 \log_{b} (9)}{2} - \frac{2 \log_{b} (27)}{3}$

Step 2

Multiply each term in $\log_{b} (x) = \frac{3 \log_{b} (9)}{2} - \frac{2 \log_{b} (27)}{3}$ by $6$ to eliminate the fractions.

Tap for more steps...

Step 2.1

Multiply each term in $\log_{b} (x) = \frac{3 \log_{b} (9)}{2} - \frac{2 \log_{b} (27)}{3}$ by $6$ .

$\log_{b} (x) \cdot 6 = \frac{3 \log_{b} (9)}{2} \cdot 6 - \frac{2 \log_{b} (27)}{3} \cdot 6$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Move $6$ to the left of $\log_{b} (x)$ .

$6 \log_{b} (x) = \frac{3 \log_{b} (9)}{2} \cdot 6 - \frac{2 \log_{b} (27)}{3} \cdot 6$

Step 2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1

Simplify each term.

Tap for more steps...

Step 2.3.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.3.1.1.1

Factor $2$ out of $6$ .

$6 \log_{b} (x) = \frac{3 \log_{b} (9)}{2} \cdot (2 (3)) - \frac{2 \log_{b} (27)}{3} \cdot 6$

Step 2.3.1.1.2

Cancel the common factor.

$6 \log_{b} (x) = \frac{3 \log_{b} (9)}{2} \cdot (2 \cdot 3) - \frac{2 \log_{b} (27)}{3} \cdot 6$

Step 2.3.1.1.3

Rewrite the expression.

$6 \log_{b} (x) = 3 \log_{b} (9) \cdot 3 - \frac{2 \log_{b} (27)}{3} \cdot 6$

Step 2.3.1.2

Multiply $3$ by $3$ .

$6 \log_{b} (x) = 9 \log_{b} (9) - \frac{2 \log_{b} (27)}{3} \cdot 6$

Step 2.3.1.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.3.1.3.1

Move the leading negative in $- \frac{2 \log_{b} (27)}{3}$ into the numerator.

$6 \log_{b} (x) = 9 \log_{b} (9) + \frac{- 2 \log_{b} (27)}{3} \cdot 6$

Step 2.3.1.3.2

Factor $3$ out of $6$ .

$6 \log_{b} (x) = 9 \log_{b} (9) + \frac{- 2 \log_{b} (27)}{3} \cdot (3 (2))$

Step 2.3.1.3.3

Cancel the common factor.

$6 \log_{b} (x) = 9 \log_{b} (9) + \frac{- 2 \log_{b} (27)}{3} \cdot (3 \cdot 2)$

Step 2.3.1.3.4

Rewrite the expression.

$6 \log_{b} (x) = 9 \log_{b} (9) - 2 \log_{b} (27) \cdot 2$

Step 2.3.1.4

Multiply $2$ by $- 2$ .

$6 \log_{b} (x) = 9 \log_{b} (9) - 4 \log_{b} (27)$

Step 3

Simplify the left side.

Tap for more steps...

Step 3.1

Simplify $6 \log_{b} (x)$ by moving $6$ inside the logarithm.

$\log_{b} (x^{6}) = 9 \log_{b} (9) - 4 \log_{b} (27)$

Step 4

Simplify the right side.

Tap for more steps...

Step 4.1

Simplify $9 \log_{b} (9) - 4 \log_{b} (27)$ .

Tap for more steps...

Step 4.1.1

Simplify each term.

Tap for more steps...

Step 4.1.1.1

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

$\log_{b} (x^{6}) = \log_{b} (9^{9}) - 4 \log_{b} (27)$

Step 4.1.1.2

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

$\log_{b} (x^{6}) = \log_{b} (387420489) - 4 \log_{b} (27)$

Step 4.1.1.3

Simplify $- 4 \log_{b} (27)$ by moving $4$ inside the logarithm.

$\log_{b} (x^{6}) = \log_{b} (387420489) - \log_{b} (27^{4})$

Step 4.1.1.4

Raise $27$ to the power of $4$ .

$\log_{b} (x^{6}) = \log_{b} (387420489) - \log_{b} (531441)$

Step 4.1.2

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

$\log_{b} (x^{6}) = \log_{b} (\frac{387420489}{531441})$

Step 4.1.3

Divide $387420489$ by $531441$ .

$\log_{b} (x^{6}) = \log_{b} (729)$

Step 5

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

$x^{6} = 729$

Step 6

Solve for $x$ .

Tap for more steps...

Step 6.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt[6]{729}$

Step 6.2

Simplify $\pm \sqrt[6]{729}$ .

Tap for more steps...

Step 6.2.1

Rewrite $729$ as $3^{6}$ .

$x = \pm \sqrt[6]{3^{6}}$

Step 6.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 3$

Step 6.3

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 6.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 3$

Step 6.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 3$

Step 6.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 3, - 3$