Algebra Examples

$\log_{5} (x^{10}) - \log_{5} (x^{6}) = 21$

Step 1

Simplify the left side.

Tap for more steps...

Step 1.1

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

$\log_{5} (\frac{x^{10}}{x^{6}}) = 21$

Step 1.2

Cancel the common factor of $x^{10}$ and $x^{6}$ .

Tap for more steps...

Step 1.2.1

Factor $x^{6}$ out of $x^{10}$ .

$\log_{5} (\frac{x^{6} x^{4}}{x^{6}}) = 21$

Step 1.2.2

Cancel the common factors.

Tap for more steps...

Step 1.2.2.1

Multiply by $1$ .

$\log_{5} (\frac{x^{6} x^{4}}{x^{6} \cdot 1}) = 21$

Step 1.2.2.2

Cancel the common factor.

$\log_{5} (\frac{x^{6} x^{4}}{x^{6} \cdot 1}) = 21$

Step 1.2.2.3

Rewrite the expression.

$\log_{5} (\frac{x^{4}}{1}) = 21$

Step 1.2.2.4

Divide $x^{4}$ by $1$ .

$\log_{5} (x^{4}) = 21$

Step 2

Write in exponential form.

Tap for more steps...

Step 2.1

For logarithmic equations, $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ such that $x > 0$ , $b > 0$ , and $b \neq 1$ . In this case, $b = 5$ , $x = x^{4}$ , and $y = 21$ .

$b = 5$

$x = x^{4}$

$y = 21$

Step 2.2

Substitute the values of $b$ , $x$ , and $y$ into the equation $b^{y} = x$ .

$5^{21} = x^{4}$

Step 3

Solve for $x$ .

Tap for more steps...

Step 3.1

Rewrite the equation as $x^{4} = 5^{21}$ .

$x^{4} = 5^{21}$

Step 3.2

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

$x = \pm \sqrt[4]{5^{21}}$

Step 3.3

Simplify $\pm \sqrt[4]{5^{21}}$ .

Tap for more steps...

Step 3.3.1

Raise $5$ to the power of $21$ .

$x = \pm \sqrt[4]{476837158203125}$

Step 3.3.2

Rewrite $476837158203125$ as $3125^{4} \cdot 5$ .

Tap for more steps...

Step 3.3.2.1

Factor $95367431640625$ out of $476837158203125$ .

$x = \pm \sqrt[4]{95367431640625 (5)}$

Step 3.3.2.2

Rewrite $95367431640625$ as $3125^{4}$ .

$x = \pm \sqrt[4]{3125^{4} \cdot 5}$

Step 3.3.3

Pull terms out from under the radical.

$x = \pm 3125 \sqrt[4]{5}$

Step 3.4

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

Tap for more steps...

Step 3.4.1

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

$x = 3125 \sqrt[4]{5}$

Step 3.4.2

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

$x = - 3125 \sqrt[4]{5}$

Step 3.4.3

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

$x = 3125 \sqrt[4]{5}, - 3125 \sqrt[4]{5}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = 3125 \sqrt[4]{5}, - 3125 \sqrt[4]{5}$

Decimal Form:

$x = 4672.96494131 \dots, - 4672.96494131 \dots$