Precalculus Examples

$1 - 6 \log (6 x - 7) = 13$

Step 1

Move all terms not containing $\log (6 x - 7)$ to the right side of the equation.

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Step 1.1

Subtract $1$ from both sides of the equation.

$- 6 \log (6 x - 7) = 13 - 1$

Step 1.2

Subtract $1$ from $13$ .

$- 6 \log (6 x - 7) = 12$

Step 2

Divide each term in $- 6 \log (6 x - 7) = 12$ by $- 6$ and simplify.

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Step 2.1

Divide each term in $- 6 \log (6 x - 7) = 12$ by $- 6$ .

$\frac{- 6 \log (6 x - 7)}{- 6} = \frac{12}{- 6}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $- 6$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{- 6 \log (6 x - 7)}{- 6} = \frac{12}{- 6}$

Step 2.2.1.2

Divide $\log (6 x - 7)$ by $1$ .

$\log (6 x - 7) = \frac{12}{- 6}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $12$ by $- 6$ .

$\log (6 x - 7) = - 2$

Step 3

Rewrite $\log (6 x - 7) = - 2$ 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 $b^{y} = x$ .

$10^{- 2} = 6 x - 7$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $6 x - 7 = 10^{- 2}$ .

$6 x - 7 = 10^{- 2}$

Step 4.2

Simplify $10^{- 2}$ .

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Step 4.2.1

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

$6 x - 7 = \frac{1}{10^{2}}$

Step 4.2.2

Raise $10$ to the power of $2$ .

$6 x - 7 = \frac{1}{100}$

Step 4.3

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

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Step 4.3.1

Add $7$ to both sides of the equation.

$6 x = \frac{1}{100} + 7$

Step 4.3.2

To write $7$ as a fraction with a common denominator, multiply by $\frac{100}{100}$ .

$6 x = \frac{1}{100} + 7 \cdot \frac{100}{100}$

Step 4.3.3

Combine $7$ and $\frac{100}{100}$ .

$6 x = \frac{1}{100} + \frac{7 \cdot 100}{100}$

Step 4.3.4

Combine the numerators over the common denominator.

$6 x = \frac{1 + 7 \cdot 100}{100}$

Step 4.3.5

Simplify the numerator.

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Step 4.3.5.1

Multiply $7$ by $100$ .

$6 x = \frac{1 + 700}{100}$

Step 4.3.5.2

Add $1$ and $700$ .

$6 x = \frac{701}{100}$

Step 4.4

Divide each term in $6 x = \frac{701}{100}$ by $6$ and simplify.

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Step 4.4.1

Divide each term in $6 x = \frac{701}{100}$ by $6$ .

$\frac{6 x}{6} = \frac{\frac{701}{100}}{6}$

Step 4.4.2

Simplify the left side.

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Step 4.4.2.1

Cancel the common factor of $6$ .

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Step 4.4.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{\frac{701}{100}}{6}$

Step 4.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{701}{100}}{6}$

Step 4.4.3

Simplify the right side.

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Step 4.4.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{701}{100} \cdot \frac{1}{6}$

Step 4.4.3.2

Multiply $\frac{701}{100} \cdot \frac{1}{6}$ .

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Step 4.4.3.2.1

Multiply $\frac{701}{100}$ by $\frac{1}{6}$ .

$x = \frac{701}{100 \cdot 6}$

Step 4.4.3.2.2

Multiply $100$ by $6$ .

$x = \frac{701}{600}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{701}{600}$

Decimal Form:

$x = 1.168\overline{3}$

Mixed Number Form:

$x = 1 \frac{101}{600}$