Algebra Examples

$\log (2 x) + \log (12) = 3$

Step 1

Simplify the left side.

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Step 1.1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log (2 x \cdot 12) = 3$

Step 1.2

Multiply $12$ by $2$ .

$\log (24 x) = 3$

Step 2

Rewrite $\log (24 x) = 3$ 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 $b^{y} = x$ .

$10^{3} = 24 x$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $24 x = 10^{3}$ .

$24 x = 10^{3}$

Step 3.2

Divide each term in $24 x = 10^{3}$ by $24$ and simplify.

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Step 3.2.1

Divide each term in $24 x = 10^{3}$ by $24$ .

$\frac{24 x}{24} = \frac{10^{3}}{24}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $24$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{24 x}{24} = \frac{10^{3}}{24}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{10^{3}}{24}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Raise $10$ to the power of $3$ .

$x = \frac{1000}{24}$

Step 3.2.3.2

Cancel the common factor of $1000$ and $24$ .

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Step 3.2.3.2.1

Factor $8$ out of $1000$ .

$x = \frac{8 (125)}{24}$

Step 3.2.3.2.2

Cancel the common factors.

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Step 3.2.3.2.2.1

Factor $8$ out of $24$ .

$x = \frac{8 \cdot 125}{8 \cdot 3}$

Step 3.2.3.2.2.2

Cancel the common factor.

$x = \frac{8 \cdot 125}{8 \cdot 3}$

Step 3.2.3.2.2.3

Rewrite the expression.

$x = \frac{125}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{125}{3}$

Decimal Form:

$x = 41.\overline{6}$

Mixed Number Form:

$x = 41 \frac{2}{3}$