Calculus Examples

$\ln (6) + \ln (x - 1) = 0$

Step 1

Simplify the left side.

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Step 1.1

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

$\ln (6 (x - 1)) = 0$

Step 1.2

Apply the distributive property.

$\ln (6 x + 6 \cdot - 1) = 0$

Step 1.3

Multiply $6$ by $- 1$ .

$\ln (6 x - 6) = 0$

Step 2

To solve for $x$ , rewrite the equation using properties of logarithms.

$e^{\ln (6 x - 6)} = e^{0}$

Step 3

Rewrite $\ln (6 x - 6) = 0$ 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 $b^{y} = x$ .

$e^{0} = 6 x - 6$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $6 x - 6 = e^{0}$ .

$6 x - 6 = e^{0}$

Step 4.2

Anything raised to $0$ is $1$ .

$6 x - 6 = 1$

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 $6$ to both sides of the equation.

$6 x = 1 + 6$

Step 4.3.2

Add $1$ and $6$ .

$6 x = 7$

Step 4.4

Divide each term in $6 x = 7$ by $6$ and simplify.

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Step 4.4.1

Divide each term in $6 x = 7$ by $6$ .

$\frac{6 x}{6} = \frac{7}{6}$

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.

$\frac{6 x}{6} = \frac{7}{6}$

Step 4.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{7}{6}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{7}{6}$

Decimal Form:

$x = 1.1\overline{6}$

Mixed Number Form:

$x = 1 \frac{1}{6}$