Solve for x log base 3 of 4x^2-1- log base 3 of x+3=1

$\log_{3} (4 x^{2} - 1) - \log_{3} (x + 3) = 1$

Step 1

Simplify the left side.

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{3} (\frac{4 x^{2} - 1}{x + 3}) = 1$

Simplify the numerator.

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

$\log_{3} (\frac{{(2 x)}^{2} - 1}{x + 3}) = 1$

Rewrite $1$ as $1^{2}$ .

$\log_{3} (\frac{{(2 x)}^{2} - 1^{2}}{x + 3}) = 1$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 x$ and $b = 1$ .

$\log_{3} (\frac{(2 x + 1) (2 x - 1)}{x + 3}) = 1$

Step 2

Rewrite $\log_{3} (\frac{(2 x + 1) (2 x - 1)}{x + 3}) = 1$ 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 $b^{y} = x$ .

$3^{1} = \frac{(2 x + 1) (2 x - 1)}{x + 3}$

Step 3

Cross multiply to remove the fraction.

$(2 x + 1) (2 x - 1) = 3 (x + 3)$

Step 4

Simplify $3 (x + 3)$ .

Apply the distributive property.

$(2 x + 1) (2 x - 1) = 3 x + 3 \cdot 3$

Multiply $3$ by $3$ .

$(2 x + 1) (2 x - 1) = 3 x + 9$

Step 5

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

Subtract $3 x$ from both sides of the equation.

$(2 x + 1) (2 x - 1) - 3 x = 9$

Simplify each term.

Expand $(2 x + 1) (2 x - 1)$ using the FOIL Method.

Apply the distributive property.

$2 x (2 x - 1) + 1 (2 x - 1) - 3 x = 9$

Apply the distributive property.

$2 x (2 x) + 2 x \cdot - 1 + 1 (2 x - 1) - 3 x = 9$

Apply the distributive property.

$2 x (2 x) + 2 x \cdot - 1 + 1 (2 x) + 1 \cdot - 1 - 3 x = 9$

Combine the opposite terms in $2 x (2 x) + 2 x \cdot - 1 + 1 (2 x) + 1 \cdot - 1$ .

Reorder the factors in the terms $2 x \cdot - 1$ and $1 (2 x)$ .

$2 x (2 x) - 1 \cdot 2 x + 1 \cdot 2 x + 1 \cdot - 1 - 3 x = 9$

Add $- 1 \cdot 2 x$ and $1 \cdot 2 x$ .

$2 x (2 x) + 0 + 1 \cdot - 1 - 3 x = 9$

Add $2 x (2 x)$ and $0$ .

$2 x (2 x) + 1 \cdot - 1 - 3 x = 9$

Simplify each term.

Rewrite using the commutative property of multiplication.

$2 \cdot 2 x \cdot x + 1 \cdot - 1 - 3 x = 9$

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

Move $x$ .

$2 \cdot 2 (x \cdot x) + 1 \cdot - 1 - 3 x = 9$

Multiply $x$ by $x$ .

$2 \cdot 2 x^{2} + 1 \cdot - 1 - 3 x = 9$

Multiply $2$ by $2$ .

$4 x^{2} + 1 \cdot - 1 - 3 x = 9$

Multiply $- 1$ by $1$ .

$4 x^{2} - 1 - 3 x = 9$

Step 6

Factor by grouping.

Reorder terms.

$4 x^{2} - 3 x - 1 = 9$

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

Factor $- 3$ out of $- 3 x$ .

$4 x^{2} - 3 x - 1 = 9$

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

$4 x^{2} + (1 - 4) x - 1 = 9$

Apply the distributive property.

$4 x^{2} + 1 x - 4 x - 1 = 9$

Multiply $x$ by $1$ .

$4 x^{2} + x - 4 x - 1 = 9$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(4 x^{2} + x) - 4 x - 1 = 9$

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

$x (4 x + 1) - (4 x + 1) = 9$

Factor the polynomial by factoring out the greatest common factor, $4 x + 1$ .

$(4 x + 1) (x - 1) = 9$

Step 7

Simplify $(4 x + 1) (x - 1)$ .

Expand $(4 x + 1) (x - 1)$ using the FOIL Method.

Apply the distributive property.

$4 x (x - 1) + 1 (x - 1) = 9$

Apply the distributive property.

$4 x \cdot x + 4 x \cdot - 1 + 1 (x - 1) = 9$

Apply the distributive property.

$4 x \cdot x + 4 x \cdot - 1 + 1 x + 1 \cdot - 1 = 9$

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

$4 (x \cdot x) + 4 x \cdot - 1 + 1 x + 1 \cdot - 1 = 9$

Multiply $x$ by $x$ .

$4 x^{2} + 4 x \cdot - 1 + 1 x + 1 \cdot - 1 = 9$

Multiply $- 1$ by $4$ .

$4 x^{2} - 4 x + 1 x + 1 \cdot - 1 = 9$

Multiply $x$ by $1$ .

$4 x^{2} - 4 x + x + 1 \cdot - 1 = 9$

Multiply $- 1$ by $1$ .

$4 x^{2} - 4 x + x - 1 = 9$

Add $- 4 x$ and $x$ .

$4 x^{2} - 3 x - 1 = 9$

Step 8

Subtract $9$ from both sides of the equation.

$4 x^{2} - 3 x - 1 - 9 = 0$

Step 9

Subtract $9$ from $- 1$ .

$4 x^{2} - 3 x - 10 = 0$

Step 10

Factor by grouping.

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 $a \cdot c = 4 \cdot - 10 = - 40$ and whose sum is $b = - 3$ .

Factor $- 3$ out of $- 3 x$ .

$4 x^{2} - 3 x - 10 = 0$

Rewrite $- 3$ as $5$ plus $- 8$

$4 x^{2} + (5 - 8) x - 10 = 0$

Apply the distributive property.

$4 x^{2} + 5 x - 8 x - 10 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(4 x^{2} + 5 x) - 8 x - 10 = 0$

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

$x (4 x + 5) - 2 (4 x + 5) = 0$

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

$(4 x + 5) (x - 2) = 0$

Step 11

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

$4 x + 5 = 0$

$x - 2 = 0$

Step 12

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

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

$4 x + 5 = 0$

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

Subtract $5$ from both sides of the equation.

$4 x = - 5$

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

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

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

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Step 13

Set $x - 2$ equal to $0$ and solve for $x$ .

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Step 14

The final solution is all the values that make $(4 x + 5) (x - 2) = 0$ true.

$x = - \frac{5}{4}, 2$

Step 15

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{5}{4}, 2$

Decimal Form:

$x = - 1.25, 2$

Mixed Number Form:

$x = - 1 \frac{1}{4}, 2$

Solve for x log base 3 of 4x^2-1- log base 3 of x+3=1

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