Finite Math Examples

$\frac{1}{2} \cdot \log_{5} (81) = \log_{5} (3 x)$

Step 1

Rewrite the equation as $\log_{5} (3 x) = \frac{1}{2} \cdot \log_{5} (81)$ .

$\log_{5} (3 x) = \frac{1}{2} \cdot \log_{5} (81)$

Step 2

Simplify the right side.

Tap for more steps...

Step 2.1

Combine $\frac{1}{2}$ and $\log_{5} (81)$ .

$\log_{5} (3 x) = \frac{\log_{5} (81)}{2}$

Step 3

Move all the terms containing a logarithm to the left side of the equation.

$\log_{5} (3 x) - \frac{\log_{5} (81)}{2} = 0$

Step 4

Simplify each term.

Tap for more steps...

Step 4.1

Rewrite $\log_{5} (81)$ as $\log_{5} (3^{4})$ .

$\log_{5} (3 x) - \frac{\log_{5} (3^{4})}{2} = 0$

Step 4.2

Expand $\log_{5} (3^{4})$ by moving $4$ outside the logarithm.

$\log_{5} (3 x) - \frac{4 \log_{5} (3)}{2} = 0$

Step 4.3

Cancel the common factor of $4$ and $2$ .

Tap for more steps...

Step 4.3.1

Factor $2$ out of $4 \log_{5} (3)$ .

$\log_{5} (3 x) - \frac{2 (2 \log_{5} (3))}{2} = 0$

Step 4.3.2

Cancel the common factors.

Tap for more steps...

Step 4.3.2.1

Factor $2$ out of $2$ .

$\log_{5} (3 x) - \frac{2 (2 \log_{5} (3))}{2 (1)} = 0$

Step 4.3.2.2

Cancel the common factor.

$\log_{5} (3 x) - \frac{2 (2 \log_{5} (3))}{2 \cdot 1} = 0$

Step 4.3.2.3

Rewrite the expression.

$\log_{5} (3 x) - \frac{2 \log_{5} (3)}{1} = 0$

Step 4.3.2.4

Divide $2 \log_{5} (3)$ by $1$ .

$\log_{5} (3 x) - (2 \log_{5} (3)) = 0$

Step 4.4

Multiply $2$ by $- 1$ .

$\log_{5} (3 x) - 2 \log_{5} (3) = 0$

Step 5

Simplify the left side.

Tap for more steps...

Step 5.1

Simplify $\log_{5} (3 x) - 2 \log_{5} (3)$ .

Tap for more steps...

Step 5.1.1

Simplify each term.

Tap for more steps...

Step 5.1.1.1

Simplify $- 2 \log_{5} (3)$ by moving $2$ inside the logarithm.

$\log_{5} (3 x) - \log_{5} (3^{2}) = 0$

Step 5.1.1.2

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

$\log_{5} (3 x) - \log_{5} (9) = 0$

Step 5.1.2

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

$\log_{5} (\frac{3 x}{9}) = 0$

Step 5.1.3

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

Tap for more steps...

Step 5.1.3.1

Factor $3$ out of $3 x$ .

$\log_{5} (\frac{3 (x)}{9}) = 0$

Step 5.1.3.2

Cancel the common factors.

Tap for more steps...

Step 5.1.3.2.1

Factor $3$ out of $9$ .

$\log_{5} (\frac{3 x}{3 \cdot 3}) = 0$

Step 5.1.3.2.2

Cancel the common factor.

$\log_{5} (\frac{3 x}{3 \cdot 3}) = 0$

Step 5.1.3.2.3

Rewrite the expression.

$\log_{5} (\frac{x}{3}) = 0$

Step 6

Rewrite $\log_{5} (\frac{x}{3}) = 0$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$5^{0} = \frac{x}{3}$

Step 7

Solve for $x$ .

Tap for more steps...

Step 7.1

Rewrite the equation as $\frac{x}{3} = 5^{0}$ .

$\frac{x}{3} = 5^{0}$

Step 7.2

Multiply both sides of the equation by $3$ .

$3 \frac{x}{3} = 3 \cdot 5^{0}$

Step 7.3

Simplify both sides of the equation.

Tap for more steps...

Step 7.3.1

Simplify the left side.

Tap for more steps...

Step 7.3.1.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 7.3.1.1.1

Cancel the common factor.

$3 \frac{x}{3} = 3 \cdot 5^{0}$

Step 7.3.1.1.2

Rewrite the expression.

$x = 3 \cdot 5^{0}$

Step 7.3.2

Simplify the right side.

Tap for more steps...

Step 7.3.2.1

Simplify $3 \cdot 5^{0}$ .

Tap for more steps...

Step 7.3.2.1.1

Anything raised to $0$ is $1$ .

$x = 3 \cdot 1$

Step 7.3.2.1.2

Multiply $3$ by $1$ .

$x = 3$