Algebra Examples

$\log_{5} (3 k + 12) = \frac{3}{4} \cdot (\log_{5} (405) - \log_{5} (5))$

Step 1

Simplify the right side.

Tap for more steps...

Step 1.1

Simplify $\frac{3}{4} \cdot (\log_{5} (405) - \log_{5} (5))$ .

Tap for more steps...

Step 1.1.1

Apply the distributive property.

$\log_{5} (3 k + 12) = \frac{3}{4} \log_{5} (405) + \frac{3}{4} (- \log_{5} (5))$

Step 1.1.2

Combine $\frac{3}{4}$ and $\log_{5} (405)$ .

$\log_{5} (3 k + 12) = \frac{3 \log_{5} (405)}{4} + \frac{3}{4} (- \log_{5} (5))$

Step 1.1.3

Combine $\frac{3}{4}$ and $\log_{5} (5)$ .

$\log_{5} (3 k + 12) = \frac{3 \log_{5} (405)}{4} - \frac{3 \log_{5} (5)}{4}$

Step 2

Multiply each term in $\log_{5} (3 k + 12) = \frac{3 \log_{5} (405)}{4} - \frac{3 \log_{5} (5)}{4}$ by $4$ to eliminate the fractions.

Tap for more steps...

Step 2.1

Multiply each term in $\log_{5} (3 k + 12) = \frac{3 \log_{5} (405)}{4} - \frac{3 \log_{5} (5)}{4}$ by $4$ .

$\log_{5} (3 k + 12) \cdot 4 = \frac{3 \log_{5} (405)}{4} \cdot 4 - \frac{3 \log_{5} (5)}{4} \cdot 4$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Move $4$ to the left of $\log_{5} (3 k + 12)$ .

$4 \log_{5} (3 k + 12) = \frac{3 \log_{5} (405)}{4} \cdot 4 - \frac{3 \log_{5} (5)}{4} \cdot 4$

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 $4$ .

Tap for more steps...

Step 2.3.1.1.1

Cancel the common factor.

$4 \log_{5} (3 k + 12) = \frac{3 \log_{5} (405)}{4} \cdot 4 - \frac{3 \log_{5} (5)}{4} \cdot 4$

Step 2.3.1.1.2

Rewrite the expression.

$4 \log_{5} (3 k + 12) = 3 \log_{5} (405) - \frac{3 \log_{5} (5)}{4} \cdot 4$

Step 2.3.1.2

Cancel the common factor of $4$ .

Tap for more steps...

Step 2.3.1.2.1

Move the leading negative in $- \frac{3 \log_{5} (5)}{4}$ into the numerator.

$4 \log_{5} (3 k + 12) = 3 \log_{5} (405) + \frac{- 3 \log_{5} (5)}{4} \cdot 4$

Step 2.3.1.2.2

Cancel the common factor.

$4 \log_{5} (3 k + 12) = 3 \log_{5} (405) + \frac{- 3 \log_{5} (5)}{4} \cdot 4$

Step 2.3.1.2.3

Rewrite the expression.

$4 \log_{5} (3 k + 12) = 3 \log_{5} (405) - 3 \log_{5} (5)$

Step 3

Simplify the left side.

Tap for more steps...

Step 3.1

Simplify $4 \log_{5} (3 k + 12)$ by moving $4$ inside the logarithm.

$\log_{5} ({(3 k + 12)}^{4}) = 3 \log_{5} (405) - 3 \log_{5} (5)$

Step 4

Simplify the right side.

Tap for more steps...

Step 4.1

Simplify each term.

Tap for more steps...

Step 4.1.1

Simplify $3 \log_{5} (405)$ by moving $3$ inside the logarithm.

$\log_{5} ({(3 k + 12)}^{4}) = \log_{5} (405^{3}) - 3 \log_{5} (5)$

Step 4.1.2

Raise $405$ to the power of $3$ .

$\log_{5} ({(3 k + 12)}^{4}) = \log_{5} (66430125) - 3 \log_{5} (5)$

Step 4.1.3

Logarithm base $5$ of $5$ is $1$ .

$\log_{5} ({(3 k + 12)}^{4}) = \log_{5} (66430125) - 3 \cdot 1$

Step 4.1.4

Multiply $- 3$ by $1$ .

$\log_{5} ({(3 k + 12)}^{4}) = \log_{5} (66430125) - 3$

Step 5

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

$\log_{5} ({(3 k + 12)}^{4}) - \log_{5} (66430125) = - 3$

Step 6

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

$\log_{5} (\frac{{(3 k + 12)}^{4}}{66430125}) = - 3$

Step 7

Simplify the numerator.

Tap for more steps...

Step 7.1

Factor $3$ out of $3 k + 12$ .

Tap for more steps...

Step 7.1.1

Factor $3$ out of $3 k$ .

$\log_{5} (\frac{{(3 (k) + 12)}^{4}}{66430125}) = - 3$

Step 7.1.2

Factor $3$ out of $12$ .

$\log_{5} (\frac{{(3 k + 3 \cdot 4)}^{4}}{66430125}) = - 3$

Step 7.1.3

Factor $3$ out of $3 k + 3 \cdot 4$ .

$\log_{5} (\frac{{(3 (k + 4))}^{4}}{66430125}) = - 3$

Step 7.2

Apply the product rule to $3 (k + 4)$ .

$\log_{5} (\frac{3^{4} {(k + 4)}^{4}}{66430125}) = - 3$

Step 7.3

Raise $3$ to the power of $4$ .

$\log_{5} (\frac{81 {(k + 4)}^{4}}{66430125}) = - 3$

Step 8

Cancel the common factor of $81$ and $66430125$ .

Tap for more steps...

Step 8.1

Factor $81$ out of $81 {(k + 4)}^{4}$ .

$\log_{5} (\frac{81 ({(k + 4)}^{4})}{66430125}) = - 3$

Step 8.2

Cancel the common factors.

Tap for more steps...

Step 8.2.1

Factor $81$ out of $66430125$ .

$\log_{5} (\frac{81 {(k + 4)}^{4}}{81 \cdot 820125}) = - 3$

Step 8.2.2

Cancel the common factor.

$\log_{5} (\frac{81 {(k + 4)}^{4}}{81 \cdot 820125}) = - 3$

Step 8.2.3

Rewrite the expression.

$\log_{5} (\frac{{(k + 4)}^{4}}{820125}) = - 3$

Step 9

Rewrite $\log_{5} (\frac{{(k + 4)}^{4}}{820125}) = - 3$ 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 $b^{y} = x$ .

$5^{- 3} = \frac{{(k + 4)}^{4}}{820125}$

Step 10

Solve for $k$ .

Tap for more steps...

Step 10.1

Rewrite the equation as $\frac{{(k + 4)}^{4}}{820125} = 5^{- 3}$ .

$\frac{{(k + 4)}^{4}}{820125} = 5^{- 3}$

Step 10.2

Multiply both sides of the equation by $820125$ .

$820125 \frac{{(k + 4)}^{4}}{820125} = 820125 \cdot 5^{- 3}$

Step 10.3

Simplify both sides of the equation.

Tap for more steps...

Step 10.3.1

Simplify the left side.

Tap for more steps...

Step 10.3.1.1

Cancel the common factor of $820125$ .

Tap for more steps...

Step 10.3.1.1.1

Cancel the common factor.

$820125 \frac{{(k + 4)}^{4}}{820125} = 820125 \cdot 5^{- 3}$

Step 10.3.1.1.2

Rewrite the expression.

${(k + 4)}^{4} = 820125 \cdot 5^{- 3}$

Step 10.3.2

Simplify the right side.

Tap for more steps...

Step 10.3.2.1

Simplify $820125 \cdot 5^{- 3}$ .

Tap for more steps...

Step 10.3.2.1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(k + 4)}^{4} = 820125 \cdot \frac{1}{5^{3}}$

Step 10.3.2.1.2

Raise $5$ to the power of $3$ .

${(k + 4)}^{4} = 820125 \cdot \frac{1}{125}$

Step 10.3.2.1.3

Cancel the common factor of $125$ .

Tap for more steps...

Step 10.3.2.1.3.1

Factor $125$ out of $820125$ .

${(k + 4)}^{4} = 125 (6561) \cdot \frac{1}{125}$

Step 10.3.2.1.3.2

Cancel the common factor.

${(k + 4)}^{4} = 125 \cdot 6561 \cdot \frac{1}{125}$

Step 10.3.2.1.3.3

Rewrite the expression.

${(k + 4)}^{4} = 6561$

Step 10.4

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

$k + 4 = \pm \sqrt[4]{6561}$

Step 10.5

Simplify $\pm \sqrt[4]{6561}$ .

Tap for more steps...

Step 10.5.1

Rewrite $6561$ as $9^{4}$ .

$k + 4 = \pm \sqrt[4]{9^{4}}$

Step 10.5.2

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

$k + 4 = \pm 9$

Step 10.6

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

Tap for more steps...

Step 10.6.1

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

$k + 4 = 9$

Step 10.6.2

Move all terms not containing $k$ to the right side of the equation.

Tap for more steps...

Step 10.6.2.1

Subtract $4$ from both sides of the equation.

$k = 9 - 4$

Step 10.6.2.2

Subtract $4$ from $9$ .

$k = 5$

Step 10.6.3

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

$k + 4 = - 9$

Step 10.6.4

Move all terms not containing $k$ to the right side of the equation.

Tap for more steps...

Step 10.6.4.1

Subtract $4$ from both sides of the equation.

$k = - 9 - 4$

Step 10.6.4.2

Subtract $4$ from $- 9$ .

$k = - 13$

Step 10.6.5

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

$k = 5, - 13$

Step 11

Exclude the solutions that do not make $\log_{5} (3 k + 12) = \frac{3}{4} \cdot (\log_{5} (405) - \log_{5} (5))$ true.

$k = 5$