Finite Math Examples

$3 \log_{8} (2 x + 7) + 8 = 10$

Step 1

Move all terms not containing $\log_{8} (2 x + 7)$ to the right side of the equation.

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Step 1.1

Subtract $8$ from both sides of the equation.

$3 \log_{8} (2 x + 7) = 10 - 8$

Step 1.2

Subtract $8$ from $10$ .

$3 \log_{8} (2 x + 7) = 2$

Step 2

Divide each term in $3 \log_{8} (2 x + 7) = 2$ by $3$ and simplify.

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Step 2.1

Divide each term in $3 \log_{8} (2 x + 7) = 2$ by $3$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Divide $\log_{8} (2 x + 7)$ by $1$ .

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

Step 3

Rewrite $\log_{8} (2 x + 7) = \frac{2}{3}$ 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 $b^{y} = x$ .

$8^{\frac{2}{3}} = 2 x + 7$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $2 x + 7 = 8^{\frac{2}{3}}$ .

$2 x + 7 = 8^{\frac{2}{3}}$

Step 4.2

Simplify $8^{\frac{2}{3}}$ .

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Step 4.2.1

Rewrite.

$2 x + 7 = 0 + 0 + 8^{\frac{2}{3}}$

Step 4.2.2

Simplify the expression.

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Step 4.2.2.1

Rewrite $8$ as $2^{3}$ .

$2 x + 7 = {(2^{3})}^{\frac{2}{3}}$

Step 4.2.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$2 x + 7 = 2^{3 (\frac{2}{3})}$

Step 4.2.3

Cancel the common factor of $3$ .

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Step 4.2.3.1

Cancel the common factor.

$2 x + 7 = 2^{3 (\frac{2}{3})}$

Step 4.2.3.2

Rewrite the expression.

$2 x + 7 = 2^{2}$

Step 4.2.4

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

$2 x + 7 = 4$

Step 4.3

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

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Step 4.3.1

Subtract $7$ from both sides of the equation.

$2 x = 4 - 7$

Step 4.3.2

Subtract $7$ from $4$ .

$2 x = - 3$

Step 4.4

Divide each term in $2 x = - 3$ by $2$ and simplify.

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Step 4.4.1

Divide each term in $2 x = - 3$ by $2$ .

$\frac{2 x}{2} = \frac{- 3}{2}$

Step 4.4.2

Simplify the left side.

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Step 4.4.2.1

Cancel the common factor of $2$ .

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Step 4.4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 3}{2}$

Step 4.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{2}$

Step 4.4.3

Simplify the right side.

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Step 4.4.3.1

Move the negative in front of the fraction.

$x = - \frac{3}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{3}{2}$

Decimal Form:

$x = - 1.5$

Mixed Number Form:

$x = - 1 \frac{1}{2}$