Algebra Examples

$\log (6 x - 3) = - 4$

Step 1

Rewrite $\log (6 x - 3) = - 4$ 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 $b^{y} = x$ .

$10^{- 4} = 6 x - 3$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite the equation as $6 x - 3 = 10^{- 4}$ .

$6 x - 3 = 10^{- 4}$

Step 2.2

Simplify $10^{- 4}$ .

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Step 2.2.1

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

$6 x - 3 = \frac{1}{10^{4}}$

Step 2.2.2

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

$6 x - 3 = \frac{1}{10000}$

Step 2.3

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

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Step 2.3.1

Add $3$ to both sides of the equation.

$6 x = \frac{1}{10000} + 3$

Step 2.3.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{10000}{10000}$ .

$6 x = \frac{1}{10000} + 3 \cdot \frac{10000}{10000}$

Step 2.3.3

Combine $3$ and $\frac{10000}{10000}$ .

$6 x = \frac{1}{10000} + \frac{3 \cdot 10000}{10000}$

Step 2.3.4

Combine the numerators over the common denominator.

$6 x = \frac{1 + 3 \cdot 10000}{10000}$

Step 2.3.5

Simplify the numerator.

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Step 2.3.5.1

Multiply $3$ by $10000$ .

$6 x = \frac{1 + 30000}{10000}$

Step 2.3.5.2

Add $1$ and $30000$ .

$6 x = \frac{30001}{10000}$

Step 2.4

Divide each term in $6 x = \frac{30001}{10000}$ by $6$ and simplify.

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Step 2.4.1

Divide each term in $6 x = \frac{30001}{10000}$ by $6$ .

$\frac{6 x}{6} = \frac{\frac{30001}{10000}}{6}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $6$ .

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Step 2.4.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{\frac{30001}{10000}}{6}$

Step 2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{30001}{10000}}{6}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{30001}{10000} \cdot \frac{1}{6}$

Step 2.4.3.2

Multiply $\frac{30001}{10000} \cdot \frac{1}{6}$ .

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Step 2.4.3.2.1

Multiply $\frac{30001}{10000}$ by $\frac{1}{6}$ .

$x = \frac{30001}{10000 \cdot 6}$

Step 2.4.3.2.2

Multiply $10000$ by $6$ .

$x = \frac{30001}{60000}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{30001}{60000}$

Decimal Form:

$x = 0.50001\overline{6}$