Algebra Examples

$\log_{7} (n) = \frac{2}{3} \cdot \log_{7} (8)$

Step 1

Simplify the right side.

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Step 1.1

Combine $\frac{2}{3}$ and $\log_{7} (8)$ .

$\log_{7} (n) = \frac{2 \log_{7} (8)}{3}$

Step 2

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

$\log_{7} (n) - \frac{2 \log_{7} (8)}{3} = 0$

Step 3

Simplify each term.

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Step 3.1

Rewrite $\log_{7} (8)$ as $\log_{7} (2^{3})$ .

$\log_{7} (n) - \frac{2 \log_{7} (2^{3})}{3} = 0$

Step 3.2

Expand $\log_{7} (2^{3})$ by moving $3$ outside the logarithm.

$\log_{7} (n) - \frac{2 (3 \log_{7} (2))}{3} = 0$

Step 3.3

Cancel the common factor of $3$ .

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Step 3.3.1

Cancel the common factor.

$\log_{7} (n) - \frac{2 (3 \log_{7} (2))}{3} = 0$

Step 3.3.2

Divide $2 (\log_{7} (2))$ by $1$ .

$\log_{7} (n) - (2 (\log_{7} (2))) = 0$

Step 3.4

Multiply $2$ by $- 1$ .

$\log_{7} (n) - 2 \log_{7} (2) = 0$

Step 4

Simplify the left side.

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Step 4.1

Simplify $\log_{7} (n) - 2 \log_{7} (2)$ .

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Step 4.1.1

Simplify each term.

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Step 4.1.1.1

Simplify $- 2 \log_{7} (2)$ by moving $2$ inside the logarithm.

$\log_{7} (n) - \log_{7} (2^{2}) = 0$

Step 4.1.1.2

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

$\log_{7} (n) - \log_{7} (4) = 0$

Step 4.1.2

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

$\log_{7} (\frac{n}{4}) = 0$

Step 5

Rewrite $\log_{7} (\frac{n}{4}) = 0$ 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 $b^{y} = x$ .

$7^{0} = \frac{n}{4}$

Step 6

Solve for $n$ .

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Step 6.1

Rewrite the equation as $\frac{n}{4} = 7^{0}$ .

$\frac{n}{4} = 7^{0}$

Step 6.2

Multiply both sides of the equation by $4$ .

$4 \frac{n}{4} = 4 \cdot 7^{0}$

Step 6.3

Simplify both sides of the equation.

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Step 6.3.1

Simplify the left side.

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Step 6.3.1.1

Cancel the common factor of $4$ .

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Step 6.3.1.1.1

Cancel the common factor.

$4 \frac{n}{4} = 4 \cdot 7^{0}$

Step 6.3.1.1.2

Rewrite the expression.

$n = 4 \cdot 7^{0}$

Step 6.3.2

Simplify the right side.

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Step 6.3.2.1

Simplify $4 \cdot 7^{0}$ .

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Step 6.3.2.1.1

Anything raised to $0$ is $1$ .

$n = 4 \cdot 1$

Step 6.3.2.1.2

Multiply $4$ by $1$ .

$n = 4$