Calculus Examples

$\ln (5 x + 1) + \ln (x) = \ln (4)$

Step 1

Simplify the left side.

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Step 1.1

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

$\ln ((5 x + 1) x) = \ln (4)$

Step 1.2

Apply the distributive property.

$\ln (5 x \cdot x + 1 x) = \ln (4)$

Step 1.3

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

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Step 1.3.1

Move $x$ .

$\ln (5 (x \cdot x) + 1 x) = \ln (4)$

Step 1.3.2

Multiply $x$ by $x$ .

$\ln (5 x^{2} + 1 x) = \ln (4)$

Step 1.4

Multiply $x$ by $1$ .

$\ln (5 x^{2} + x) = \ln (4)$

Step 2

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

$5 x^{2} + x = 4$

Step 3

Solve for $x$ .

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Step 3.1

Subtract $4$ from both sides of the equation.

$5 x^{2} + x - 4 = 0$

Step 3.2

Factor by grouping.

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Step 3.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 5 \cdot - 4 = - 20$ and whose sum is $b = 1$ .

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Step 3.2.1.1

Multiply by $1$ .

$5 x^{2} + 1 x - 4 = 0$

Step 3.2.1.2

Rewrite $1$ as $- 4$ plus $5$

$5 x^{2} + (- 4 + 5) x - 4 = 0$

Step 3.2.1.3

Apply the distributive property.

$5 x^{2} - 4 x + 5 x - 4 = 0$

Step 3.2.2

Factor out the greatest common factor from each group.

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Step 3.2.2.1

Group the first two terms and the last two terms.

$(5 x^{2} - 4 x) + 5 x - 4 = 0$

Step 3.2.2.2

Factor out the greatest common factor (GCF) from each group.

$x (5 x - 4) + 1 (5 x - 4) = 0$

Step 3.2.3

Factor the polynomial by factoring out the greatest common factor, $5 x - 4$ .

$(5 x - 4) (x + 1) = 0$

Step 3.3

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

$5 x - 4 = 0$

$x + 1 = 0$

Step 3.4

Set $5 x - 4$ equal to $0$ and solve for $x$ .

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Step 3.4.1

Set $5 x - 4$ equal to $0$ .

$5 x - 4 = 0$

Step 3.4.2

Solve $5 x - 4 = 0$ for $x$ .

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Step 3.4.2.1

Add $4$ to both sides of the equation.

$5 x = 4$

Step 3.4.2.2

Divide each term in $5 x = 4$ by $5$ and simplify.

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Step 3.4.2.2.1

Divide each term in $5 x = 4$ by $5$ .

$\frac{5 x}{5} = \frac{4}{5}$

Step 3.4.2.2.2

Simplify the left side.

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Step 3.4.2.2.2.1

Cancel the common factor of $5$ .

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Step 3.4.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{4}{5}$

Step 3.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{5}$

Step 3.5

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 3.5.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 3.5.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 3.6

The final solution is all the values that make $(5 x - 4) (x + 1) = 0$ true.

$x = \frac{4}{5}, - 1$

Step 4

Exclude the solutions that do not make $\ln (5 x + 1) + \ln (x) = \ln (4)$ true.

$x = \frac{4}{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{4}{5}$

Decimal Form:

$x = 0.8$