Algebra Examples

$3 \log (2 x) = 4$

Step 1

Subtract $4$ from both sides of the equation.

$3 \log (2 x) - 4 = 0$

Step 2

Simplify each term.

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Step 2.1

Simplify $3 \log (2 x)$ by moving $3$ inside the logarithm.

$\log ({(2 x)}^{3}) - 4 = 0$

Step 2.2

Apply the product rule to $2 x$ .

$\log (2^{3} x^{3}) - 4 = 0$

Step 2.3

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

$\log (8 x^{3}) - 4 = 0$

Step 3

Add $4$ to both sides of the equation.

$\log (8 x^{3}) = 4$

Step 4

Rewrite $\log (8 x^{3}) = 4$ 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 $b^{y} = x$ .

$10^{4} = 8 x^{3}$

Step 5

Solve for $x$ .

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Step 5.1

Rewrite the equation as $8 x^{3} = 10^{4}$ .

$8 x^{3} = 10^{4}$

Step 5.2

Subtract $10^{4}$ from both sides of the equation.

$8 x^{3} - 10^{4} = 0$

Step 5.3

Simplify each term.

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Step 5.3.1

Raise $10$ to the power of $4$ .

$8 x^{3} - 1 \cdot 10000 = 0$

Step 5.3.2

Multiply $- 1$ by $10000$ .

$8 x^{3} - 10000 = 0$

Step 5.4

Factor $8$ out of $8 x^{3} - 10000$ .

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Step 5.4.1

Factor $8$ out of $8 x^{3}$ .

$8 (x^{3}) - 10000 = 0$

Step 5.4.2

Factor $8$ out of $- 10000$ .

$8 x^{3} + 8 \cdot - 1250 = 0$

Step 5.4.3

Factor $8$ out of $8 x^{3} + 8 \cdot - 1250$ .

$8 (x^{3} - 1250) = 0$

Step 5.5

Divide each term in $8 (x^{3} - 1250) = 0$ by $8$ and simplify.

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Step 5.5.1

Divide each term in $8 (x^{3} - 1250) = 0$ by $8$ .

$\frac{8 (x^{3} - 1250)}{8} = \frac{0}{8}$

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $8$ .

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Step 5.5.2.1.1

Cancel the common factor.

$\frac{8 (x^{3} - 1250)}{8} = \frac{0}{8}$

Step 5.5.2.1.2

Divide $x^{3} - 1250$ by $1$ .

$x^{3} - 1250 = \frac{0}{8}$

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Divide $0$ by $8$ .

$x^{3} - 1250 = 0$

Step 5.6

Add $1250$ to both sides of the equation.

$x^{3} = 1250$

Step 5.7

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

$x = \sqrt[3]{1250}$

Step 5.8

Simplify $\sqrt[3]{1250}$ .

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Step 5.8.1

Rewrite $1250$ as $5^{3} \cdot 10$ .

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Step 5.8.1.1

Factor $125$ out of $1250$ .

$x = \sqrt[3]{125 (10)}$

Step 5.8.1.2

Rewrite $125$ as $5^{3}$ .

$x = \sqrt[3]{5^{3} \cdot 10}$

Step 5.8.2

Pull terms out from under the radical.

$x = 5 \sqrt[3]{10}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = 5 \sqrt[3]{10}$

Decimal Form:

$x = 10.77217345 \dots$