Calculus Examples

$\ln (5 x - 6) + \ln (6 x) = 2 \ln (x)$

Step 1

Simplify the left side.

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Step 1.1

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

$\ln ((5 x - 6) (6 x)) = 2 \ln (x)$

Step 1.2

Simplify by multiplying through.

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Step 1.2.1

Apply the distributive property.

$\ln (5 x (6 x) - 6 (6 x)) = 2 \ln (x)$

Step 1.2.2

Simplify the expression.

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Step 1.2.2.1

Rewrite using the commutative property of multiplication.

$\ln (5 \cdot (6 x \cdot x) - 6 (6 x)) = 2 \ln (x)$

Step 1.2.2.2

Multiply $6$ by $- 6$ .

$\ln (5 \cdot (6 x \cdot x) - 36 x) = 2 \ln (x)$

Step 1.3

Simplify each term.

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Step 1.3.1

Multiply $x$ by $x$ by adding the exponents.

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Step 1.3.1.1

Move $x$ .

$\ln (5 \cdot (6 (x \cdot x)) - 36 x) = 2 \ln (x)$

Step 1.3.1.2

Multiply $x$ by $x$ .

$\ln (5 \cdot (6 x^{2}) - 36 x) = 2 \ln (x)$

Step 1.3.2

Multiply $5$ by $6$ .

$\ln (30 x^{2} - 36 x) = 2 \ln (x)$

Step 2

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

$\ln (30 x^{2} - 36 x) = \ln (x^{2})$

Step 3

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$30 x^{2} - 36 x = x^{2}$

Step 4

Solve for $x$ .

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Step 4.1

Move all terms containing $x$ to the left side of the equation.

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Step 4.1.1

Subtract $x^{2}$ from both sides of the equation.

$30 x^{2} - 36 x - x^{2} = 0$

Step 4.1.2

Subtract $x^{2}$ from $30 x^{2}$ .

$29 x^{2} - 36 x = 0$

Step 4.2

Factor $x$ out of $29 x^{2} - 36 x$ .

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Step 4.2.1

Factor $x$ out of $29 x^{2}$ .

$x (29 x) - 36 x = 0$

Step 4.2.2

Factor $x$ out of $- 36 x$ .

$x (29 x) + x \cdot - 36 = 0$

Step 4.2.3

Factor $x$ out of $x (29 x) + x \cdot - 36$ .

$x (29 x - 36) = 0$

Step 4.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$29 x - 36 = 0$

Step 4.4

Set $x$ equal to $0$ .

$x = 0$

Step 4.5

Set $29 x - 36$ equal to $0$ and solve for $x$ .

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Step 4.5.1

Set $29 x - 36$ equal to $0$ .

$29 x - 36 = 0$

Step 4.5.2

Solve $29 x - 36 = 0$ for $x$ .

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Step 4.5.2.1

Add $36$ to both sides of the equation.

$29 x = 36$

Step 4.5.2.2

Divide each term in $29 x = 36$ by $29$ and simplify.

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Step 4.5.2.2.1

Divide each term in $29 x = 36$ by $29$ .

$\frac{29 x}{29} = \frac{36}{29}$

Step 4.5.2.2.2

Simplify the left side.

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Step 4.5.2.2.2.1

Cancel the common factor of $29$ .

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Step 4.5.2.2.2.1.1

Cancel the common factor.

$\frac{29 x}{29} = \frac{36}{29}$

Step 4.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{36}{29}$

Step 4.6

The final solution is all the values that make $x (29 x - 36) = 0$ true.

$x = 0, \frac{36}{29}$

Step 5

Exclude the solutions that do not make $\ln (5 x - 6) + \ln (6 x) = 2 \ln (x)$ true.

$x = \frac{36}{29}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{36}{29}$

Decimal Form:

$x = 1.24137931 \dots$

Mixed Number Form:

$x = 1 \frac{7}{29}$