Solve for x log base x of 8x-3- log base x of 4=2

$\log_{x} (8 x - 3) - \log_{x} (4) = 2$

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

$\log_{x} (\frac{8 x - 3}{4}) = 2$

Rewrite $\log_{x} (\frac{8 x - 3}{4}) = 2$ 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 $b^{y} = x$ .

$x^{2} = \frac{8 x - 3}{4}$

Solve for $x$ .

Multiply both sides by $4$ .

$x^{2} \cdot 4 = \frac{8 x - 3}{4} \cdot 4$

Simplify.

Simplify the left side.

Move $4$ to the left of $x^{2}$ .

$4 x^{2} = \frac{8 x - 3}{4} \cdot 4$

Simplify the right side.

Cancel the common factor of $4$ .

Cancel the common factor.

$4 x^{2} = \frac{8 x - 3}{4} \cdot 4$

Rewrite the expression.

$4 x^{2} = 8 x - 3$

Solve for $x$ .

Subtract $8 x$ from both sides of the equation.

$4 x^{2} - 8 x = - 3$

Add $3$ to both sides of the equation.

$4 x^{2} - 8 x + 3 = 0$

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 $a \cdot c = 4 \cdot 3 = 12$ and whose sum is $b = - 8$ .

Factor $- 8$ out of $- 8 x$ .

$4 x^{2} - 8 x + 3 = 0$

Rewrite $- 8$ as $- 2$ plus $- 6$

$4 x^{2} + (- 2 - 6) x + 3 = 0$

Apply the distributive property.

$4 x^{2} - 2 x - 6 x + 3 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(4 x^{2} - 2 x) - 6 x + 3 = 0$

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

$2 x (2 x - 1) - 3 (2 x - 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

$(2 x - 1) (2 x - 3) = 0$

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

$2 x - 1 = 0$

$2 x - 3 = 0$

Set $2 x - 1$ equal to $0$ and solve for $x$ .

Set $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Solve $2 x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$2 x = 1$

Divide each term in $2 x = 1$ by $2$ and simplify.

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Divide $x$ by $1$ .

$x = \frac{1}{2}$

Set $2 x - 3$ equal to $0$ and solve for $x$ .

Set $2 x - 3$ equal to $0$ .

$2 x - 3 = 0$

Solve $2 x - 3 = 0$ for $x$ .

Add $3$ to both sides of the equation.

$2 x = 3$

Divide each term in $2 x = 3$ by $2$ and simplify.

Divide each term in $2 x = 3$ by $2$ .

$\frac{2 x}{2} = \frac{3}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{3}{2}$

Divide $x$ by $1$ .

$x = \frac{3}{2}$

The final solution is all the values that make $(2 x - 1) (2 x - 3) = 0$ true.

$x = \frac{1}{2}, \frac{3}{2}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{2}, \frac{3}{2}$

Decimal Form:

$x = 0.5, 1.5$

Mixed Number Form:

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

Solve for x log base x of 8x-3- log base x of 4=2

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