Algebra Examples

$2 \log (m - 1) + \log (2) = \log (2 m^{2} - 1)$

Step 1

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

$2 \log (m - 1) + \log (2) - \log (2 m^{2} - 1) = 0$

Step 2

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

$2 \log (m - 1) + \log (\frac{2}{2 m^{2} - 1}) = 0$

Step 3

Simplify the left side.

Tap for more steps...

Step 3.1

Simplify $2 \log (m - 1) + \log (\frac{2}{2 m^{2} - 1})$ .

Tap for more steps...

Step 3.1.1

Simplify $2 \log (m - 1)$ by moving $2$ inside the logarithm.

$\log ({(m - 1)}^{2}) + \log (\frac{2}{2 m^{2} - 1}) = 0$

Step 3.1.2

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

$\log ({(m - 1)}^{2} \frac{2}{2 m^{2} - 1}) = 0$

Step 3.1.3

Combine ${(m - 1)}^{2}$ and $\frac{2}{2 m^{2} - 1}$ .

$\log (\frac{{(m - 1)}^{2} \cdot 2}{2 m^{2} - 1}) = 0$

Step 3.1.4

Move $2$ to the left of ${(m - 1)}^{2}$ .

$\log (\frac{2 {(m - 1)}^{2}}{2 m^{2} - 1}) = 0$

Step 4

Rewrite $\log (\frac{2 {(m - 1)}^{2}}{2 m^{2} - 1}) = 0$ 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 $b^{y} = x$ .

$10^{0} = \frac{2 {(m - 1)}^{2}}{2 m^{2} - 1}$

Step 5

Cross multiply to remove the fraction.

$2 {(m - 1)}^{2} = 10^{0} (2 m^{2} - 1)$

Step 6

Simplify $10^{0} (2 m^{2} - 1)$ .

Tap for more steps...

Step 6.1

Anything raised to $0$ is $1$ .

$2 {(m - 1)}^{2} = 1 (2 m^{2} - 1)$

Step 6.2

Multiply $2 m^{2} - 1$ by $1$ .

$2 {(m - 1)}^{2} = 2 m^{2} - 1$

Step 7

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

Tap for more steps...

Step 7.1

Subtract $2 m^{2}$ from both sides of the equation.

$2 {(m - 1)}^{2} - 2 m^{2} = - 1$

Step 7.2

Simplify each term.

Tap for more steps...

Step 7.2.1

Rewrite ${(m - 1)}^{2}$ as $(m - 1) (m - 1)$ .

$2 ((m - 1) (m - 1)) - 2 m^{2} = - 1$

Step 7.2.2

Expand $(m - 1) (m - 1)$ using the FOIL Method.

Tap for more steps...

Step 7.2.2.1

Apply the distributive property.

$2 (m (m - 1) - 1 (m - 1)) - 2 m^{2} = - 1$

Step 7.2.2.2

Apply the distributive property.

$2 (m \cdot m + m \cdot - 1 - 1 (m - 1)) - 2 m^{2} = - 1$

Step 7.2.2.3

Apply the distributive property.

$2 (m \cdot m + m \cdot - 1 - 1 m - 1 \cdot - 1) - 2 m^{2} = - 1$

Step 7.2.3

Simplify and combine like terms.

Tap for more steps...

Step 7.2.3.1

Simplify each term.

Tap for more steps...

Step 7.2.3.1.1

Multiply $m$ by $m$ .

$2 (m^{2} + m \cdot - 1 - 1 m - 1 \cdot - 1) - 2 m^{2} = - 1$

Step 7.2.3.1.2

Move $- 1$ to the left of $m$ .

$2 (m^{2} - 1 \cdot m - 1 m - 1 \cdot - 1) - 2 m^{2} = - 1$

Step 7.2.3.1.3

Rewrite $- 1 m$ as $- m$ .

$2 (m^{2} - m - 1 m - 1 \cdot - 1) - 2 m^{2} = - 1$

Step 7.2.3.1.4

Rewrite $- 1 m$ as $- m$ .

$2 (m^{2} - m - m - 1 \cdot - 1) - 2 m^{2} = - 1$

Step 7.2.3.1.5

Multiply $- 1$ by $- 1$ .

$2 (m^{2} - m - m + 1) - 2 m^{2} = - 1$

Step 7.2.3.2

Subtract $m$ from $- m$ .

$2 (m^{2} - 2 m + 1) - 2 m^{2} = - 1$

Step 7.2.4

Apply the distributive property.

$2 m^{2} + 2 (- 2 m) + 2 \cdot 1 - 2 m^{2} = - 1$

Step 7.2.5

Simplify.

Tap for more steps...

Step 7.2.5.1

Multiply $- 2$ by $2$ .

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

Step 7.2.5.2

Multiply $2$ by $1$ .

$2 m^{2} - 4 m + 2 - 2 m^{2} = - 1$

Step 7.3

Combine the opposite terms in $2 m^{2} - 4 m + 2 - 2 m^{2}$ .

Tap for more steps...

Step 7.3.1

Subtract $2 m^{2}$ from $2 m^{2}$ .

$- 4 m + 2 + 0 = - 1$

Step 7.3.2

Add $- 4 m + 2$ and $0$ .

$- 4 m + 2 = - 1$

Step 8

Factor $2$ out of $- 4 m + 2$ .

Tap for more steps...

Step 8.1

Factor $2$ out of $- 4 m$ .

$2 (- 2 m) + 2 = - 1$

Step 8.2

Factor $2$ out of $2$ .

$2 (- 2 m) + 2 (1) = - 1$

Step 8.3

Factor $2$ out of $2 (- 2 m) + 2 (1)$ .

$2 (- 2 m + 1) = - 1$

Step 9

Divide each term in $2 (- 2 m + 1) = - 1$ by $2$ and simplify.

Tap for more steps...

Step 9.1

Divide each term in $2 (- 2 m + 1) = - 1$ by $2$ .

$\frac{2 (- 2 m + 1)}{2} = \frac{- 1}{2}$

Step 9.2

Simplify the left side.

Tap for more steps...

Step 9.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 9.2.1.1

Cancel the common factor.

$\frac{2 (- 2 m + 1)}{2} = \frac{- 1}{2}$

Step 9.2.1.2

Divide $- 2 m + 1$ by $1$ .

$- 2 m + 1 = \frac{- 1}{2}$

Step 9.3

Simplify the right side.

Tap for more steps...

Step 9.3.1

Move the negative in front of the fraction.

$- 2 m + 1 = - \frac{1}{2}$

Step 10

Move all terms not containing $m$ to the right side of the equation.

Tap for more steps...

Step 10.1

Subtract $1$ from both sides of the equation.

$- 2 m = - \frac{1}{2} - 1$

Step 10.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- 2 m = - \frac{1}{2} - 1 \cdot \frac{2}{2}$

Step 10.3

Combine $- 1$ and $\frac{2}{2}$ .

$- 2 m = - \frac{1}{2} + \frac{- 1 \cdot 2}{2}$

Step 10.4

Combine the numerators over the common denominator.

$- 2 m = \frac{- 1 - 1 \cdot 2}{2}$

Step 10.5

Simplify the numerator.

Tap for more steps...

Step 10.5.1

Multiply $- 1$ by $2$ .

$- 2 m = \frac{- 1 - 2}{2}$

Step 10.5.2

Subtract $2$ from $- 1$ .

$- 2 m = \frac{- 3}{2}$

Step 10.6

Move the negative in front of the fraction.

$- 2 m = - \frac{3}{2}$

Step 11

Divide each term in $- 2 m = - \frac{3}{2}$ by $- 2$ and simplify.

Tap for more steps...

Step 11.1

Divide each term in $- 2 m = - \frac{3}{2}$ by $- 2$ .

$\frac{- 2 m}{- 2} = \frac{- \frac{3}{2}}{- 2}$

Step 11.2

Simplify the left side.

Tap for more steps...

Step 11.2.1

Cancel the common factor of $- 2$ .

Tap for more steps...

Step 11.2.1.1

Cancel the common factor.

$\frac{- 2 m}{- 2} = \frac{- \frac{3}{2}}{- 2}$

Step 11.2.1.2

Divide $m$ by $1$ .

$m = \frac{- \frac{3}{2}}{- 2}$

Step 11.3

Simplify the right side.

Tap for more steps...

Step 11.3.1

Multiply the numerator by the reciprocal of the denominator.

$m = - \frac{3}{2} \cdot \frac{1}{- 2}$

Step 11.3.2

Move the negative in front of the fraction.

$m = - \frac{3}{2} (- \frac{1}{2})$

Step 11.3.3

Multiply $- \frac{3}{2} (- \frac{1}{2})$ .

Tap for more steps...

Step 11.3.3.1

Multiply $- 1$ by $- 1$ .

$m = 1 (\frac{3}{2}) \frac{1}{2}$

Step 11.3.3.2

Multiply $\frac{3}{2}$ by $1$ .

$m = \frac{3}{2} \cdot \frac{1}{2}$

Step 11.3.3.3

Multiply $\frac{3}{2}$ by $\frac{1}{2}$ .

$m = \frac{3}{2 \cdot 2}$

Step 11.3.3.4

Multiply $2$ by $2$ .

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

Step 12

Exclude the solutions that do not make $2 \log (m - 1) + \log (2) = \log (2 m^{2} - 1)$ true.

$No solution$