Algebra Examples

$2 \log (2) - \log (x) = \log (x + 3) - \log (7)$

Step 1

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

$2 \log (2) - \log (x) = \log (\frac{x + 3}{7})$

Step 2

Move all the terms containing a logarithm to the left side of the equation.

$2 \log (2) - \log (x) - \log (\frac{x + 3}{7}) = 0$

Step 3

Simplify the left side.

Tap for more steps...

Step 3.1

Simplify $2 \log (2) - \log (x) - \log (\frac{x + 3}{7})$ .

Tap for more steps...

Step 3.1.1

Simplify each term.

Tap for more steps...

Step 3.1.1.1

Simplify $2 \log (2)$ by moving $2$ inside the logarithm.

$\log (2^{2}) - \log (x) - \log (\frac{x + 3}{7}) = 0$

Step 3.1.1.2

Raise $2$ to the power of $2$ .

$\log (4) - \log (x) - \log (\frac{x + 3}{7}) = 0$

Step 3.1.2

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

$\log (\frac{4}{x}) - \log (\frac{x + 3}{7}) = 0$

Step 3.1.3

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

$\log (\frac{\frac{4}{x}}{\frac{x + 3}{7}}) = 0$

Step 3.1.4

Multiply the numerator by the reciprocal of the denominator.

$\log (\frac{4}{x} \cdot \frac{7}{x + 3}) = 0$

Step 3.1.5

Multiply $\frac{4}{x} \cdot \frac{7}{x + 3}$ .

Tap for more steps...

Step 3.1.5.1

Multiply $\frac{4}{x}$ by $\frac{7}{x + 3}$ .

$\log (\frac{4 \cdot 7}{x (x + 3)}) = 0$

Step 3.1.5.2

Multiply $4$ by $7$ .

$\log (\frac{28}{x (x + 3)}) = 0$

Step 4

Rewrite $\log (\frac{28}{x (x + 3)}) = 0$ 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 $b^{y} = x$ .

$10^{0} = \frac{28}{x (x + 3)}$

Step 5

Solve for $x$ .

Tap for more steps...

Step 5.1

Rewrite the equation as $\frac{28}{x (x + 3)} = 10^{0}$ .

$\frac{28}{x (x + 3)} = 10^{0}$

Step 5.2

Anything raised to $0$ is $1$ .

$\frac{28}{x (x + 3)} = 1$

Step 5.3

Find the LCD of the terms in the equation.

Tap for more steps...

Step 5.3.1

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

$x (x + 3), 1$

Step 5.3.2

The LCM of one and any expression is the expression.

$x (x + 3)$

Step 5.4

Multiply each term in $\frac{28}{x (x + 3)} = 1$ by $x (x + 3)$ to eliminate the fractions.

Tap for more steps...

Step 5.4.1

Multiply each term in $\frac{28}{x (x + 3)} = 1$ by $x (x + 3)$ .

$\frac{28}{x (x + 3)} (x (x + 3)) = 1 (x (x + 3))$

Step 5.4.2

Simplify the left side.

Tap for more steps...

Step 5.4.2.1

Cancel the common factor of $x (x + 3)$ .

Tap for more steps...

Step 5.4.2.1.1

Cancel the common factor.

$\frac{28}{x (x + 3)} (x (x + 3)) = 1 (x (x + 3))$

Step 5.4.2.1.2

Rewrite the expression.

$28 = 1 (x (x + 3))$

Step 5.4.3

Simplify the right side.

Tap for more steps...

Step 5.4.3.1

Multiply $x (x + 3)$ by $1$ .

$28 = x (x + 3)$

Step 5.4.3.2

Apply the distributive property.

$28 = x \cdot x + x \cdot 3$

Step 5.4.3.3

Simplify the expression.

Tap for more steps...

Step 5.4.3.3.1

Multiply $x$ by $x$ .

$28 = x^{2} + x \cdot 3$

Step 5.4.3.3.2

Move $3$ to the left of $x$ .

$28 = x^{2} + 3 x$

Step 5.5

Solve the equation.

Tap for more steps...

Step 5.5.1

Rewrite the equation as $x^{2} + 3 x = 28$ .

$x^{2} + 3 x = 28$

Step 5.5.2

Subtract $28$ from both sides of the equation.

$x^{2} + 3 x - 28 = 0$

Step 5.5.3

Factor $x^{2} + 3 x - 28$ using the AC method.

Tap for more steps...

Step 5.5.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 28$ and whose sum is $3$ .

$- 4, 7$

Step 5.5.3.2

Write the factored form using these integers.

$(x - 4) (x + 7) = 0$

Step 5.5.4

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

$x - 4 = 0$

$x + 7 = 0$

Step 5.5.5

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

Tap for more steps...

Step 5.5.5.1

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

$x - 4 = 0$

Step 5.5.5.2

Add $4$ to both sides of the equation.

$x = 4$

Step 5.5.6

Set $x + 7$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 5.5.6.1

Set $x + 7$ equal to $0$ .

$x + 7 = 0$

Step 5.5.6.2

Subtract $7$ from both sides of the equation.

$x = - 7$

Step 5.5.7

The final solution is all the values that make $(x - 4) (x + 7) = 0$ true.

$x = 4, - 7$

Step 6

Exclude the solutions that do not make $2 \log (2) - \log (x) = \log (x + 3) - \log (7)$ true.

$x = 4$