Calculus Examples

$\log (4 p + 1) + \log (p) = \log (3)$

Step 1

Simplify the left side.

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Step 1.1

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

$\log ((4 p + 1) p) = \log (3)$

Step 1.2

Apply the distributive property.

$\log (4 p \cdot p + 1 p) = \log (3)$

Step 1.3

Multiply $p$ by $p$ by adding the exponents.

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Step 1.3.1

Move $p$ .

$\log (4 (p \cdot p) + 1 p) = \log (3)$

Step 1.3.2

Multiply $p$ by $p$ .

$\log (4 p^{2} + 1 p) = \log (3)$

Step 1.4

Multiply $p$ by $1$ .

$\log (4 p^{2} + p) = \log (3)$

Step 2

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

$4 p^{2} + p = 3$

Step 3

Solve for $p$ .

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Step 3.1

Subtract $3$ from both sides of the equation.

$4 p^{2} + p - 3 = 0$

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 $a \cdot c = 4 \cdot - 3 = - 12$ and whose sum is $b = 1$ .

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Step 3.2.1.1

Multiply by $1$ .

$4 p^{2} + 1 p - 3 = 0$

Step 3.2.1.2

Rewrite $1$ as $- 3$ plus $4$

$4 p^{2} + (- 3 + 4) p - 3 = 0$

Step 3.2.1.3

Apply the distributive property.

$4 p^{2} - 3 p + 4 p - 3 = 0$

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.

$(4 p^{2} - 3 p) + 4 p - 3 = 0$

Step 3.2.2.2

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

$p (4 p - 3) + 1 (4 p - 3) = 0$

Step 3.2.3

Factor the polynomial by factoring out the greatest common factor, $4 p - 3$ .

$(4 p - 3) (p + 1) = 0$

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 $0$ .

$4 p - 3 = 0$

$p + 1 = 0$

Step 3.4

Set $4 p - 3$ equal to $0$ and solve for $p$ .

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Step 3.4.1

Set $4 p - 3$ equal to $0$ .

$4 p - 3 = 0$

Step 3.4.2

Solve $4 p - 3 = 0$ for $p$ .

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Step 3.4.2.1

Add $3$ to both sides of the equation.

$4 p = 3$

Step 3.4.2.2

Divide each term in $4 p = 3$ by $4$ and simplify.

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Step 3.4.2.2.1

Divide each term in $4 p = 3$ by $4$ .

$\frac{4 p}{4} = \frac{3}{4}$

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 $4$ .

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Step 3.4.2.2.2.1.1

Cancel the common factor.

$\frac{4 p}{4} = \frac{3}{4}$

Step 3.4.2.2.2.1.2

Divide $p$ by $1$ .

$p = \frac{3}{4}$

Step 3.5

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

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Step 3.5.1

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

$p + 1 = 0$

Step 3.5.2

Subtract $1$ from both sides of the equation.

$p = - 1$

Step 3.6

The final solution is all the values that make $(4 p - 3) (p + 1) = 0$ true.

$p = \frac{3}{4}, - 1$

Step 4

Exclude the solutions that do not make $\log (4 p + 1) + \log (p) = \log (3)$ true.

$p = \frac{3}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$p = \frac{3}{4}$

Decimal Form:

$p = 0.75$