Algebra Examples

$\log_{9} (\frac{10}{2 x}) = \log_{9} (2 x - 3)$

Step 1

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

$\frac{10}{2 x} = 2 x - 3$

Step 2

Solve for $x$ .

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Step 2.1

Reduce the expression $\frac{10}{2 x}$ by cancelling the common factors.

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Step 2.1.1

Factor $2$ out of $10$ .

$\frac{2 \cdot 5}{2 x} = 2 x - 3$

Step 2.1.2

Factor $2$ out of $2 x$ .

$\frac{2 \cdot 5}{2 (x)} = 2 x - 3$

Step 2.1.3

Cancel the common factor.

$\frac{2 \cdot 5}{2 x} = 2 x - 3$

Step 2.1.4

Rewrite the expression.

$\frac{5}{x} = 2 x - 3$

Step 2.2

Find the LCD of the terms in the equation.

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Step 2.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x, 1, 1$

Step 2.2.2

The LCM of one and any expression is the expression.

$x$

Step 2.3

Multiply each term in $\frac{5}{x} = 2 x - 3$ by $x$ to eliminate the fractions.

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Step 2.3.1

Multiply each term in $\frac{5}{x} = 2 x - 3$ by $x$ .

$\frac{5}{x} x = 2 x \cdot x - 3 x$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $x$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{5}{x} x = 2 x \cdot x - 3 x$

Step 2.3.2.1.2

Rewrite the expression.

$5 = 2 x \cdot x - 3 x$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

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

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Step 2.3.3.1.1

Move $x$ .

$5 = 2 (x \cdot x) - 3 x$

Step 2.3.3.1.2

Multiply $x$ by $x$ .

$5 = 2 x^{2} - 3 x$

Step 2.4

Solve the equation.

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Step 2.4.1

Rewrite the equation as $2 x^{2} - 3 x = 5$ .

$2 x^{2} - 3 x = 5$

Step 2.4.2

Subtract $5$ from both sides of the equation.

$2 x^{2} - 3 x - 5 = 0$

Step 2.4.3

Factor by grouping.

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Step 2.4.3.1

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 $a \cdot c = 2 \cdot - 5 = - 10$ and whose sum is $b = - 3$ .

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Step 2.4.3.1.1

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

$2 x^{2} - 3 x - 5 = 0$

Step 2.4.3.1.2

Rewrite $- 3$ as $2$ plus $- 5$

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

Step 2.4.3.1.3

Apply the distributive property.

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

Step 2.4.3.2

Factor out the greatest common factor from each group.

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Step 2.4.3.2.1

Group the first two terms and the last two terms.

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

Step 2.4.3.2.2

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

$2 x (x + 1) - 5 (x + 1) = 0$

Step 2.4.3.3

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

$(x + 1) (2 x - 5) = 0$

Step 2.4.4

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

$x + 1 = 0$

$2 x - 5 = 0$

Step 2.4.5

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

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Step 2.4.5.1

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

$x + 1 = 0$

Step 2.4.5.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 2.4.6

Set $2 x - 5$ equal to $0$ and solve for $x$ .

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Step 2.4.6.1

Set $2 x - 5$ equal to $0$ .

$2 x - 5 = 0$

Step 2.4.6.2

Solve $2 x - 5 = 0$ for $x$ .

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Step 2.4.6.2.1

Add $5$ to both sides of the equation.

$2 x = 5$

Step 2.4.6.2.2

Divide each term in $2 x = 5$ by $2$ and simplify.

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Step 2.4.6.2.2.1

Divide each term in $2 x = 5$ by $2$ .

$\frac{2 x}{2} = \frac{5}{2}$

Step 2.4.6.2.2.2

Simplify the left side.

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Step 2.4.6.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.4.6.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{5}{2}$

Step 2.4.6.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{2}$

Step 2.4.7

The final solution is all the values that make $(x + 1) (2 x - 5) = 0$ true.

$x = - 1, \frac{5}{2}$

Step 3

Exclude the solutions that do not make $\log_{9} (\frac{10}{2 x}) = \log_{9} (2 x - 3)$ true.

$x = \frac{5}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5}{2}$

Decimal Form:

$x = 2.5$

Mixed Number Form:

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