Finite Math Examples

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

Step 1

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

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

Step 2

Rewrite $\log (\frac{2 x - 3}{7 x}) = 2$ 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 $b^{y} = x$ .

$10^{2} = \frac{2 x - 3}{7 x}$

Step 3

Cross multiply to remove the fraction.

$2 x - 3 = 10^{2} (7 x)$

Step 4

Simplify $10^{2} (7 x)$ .

Tap for more steps...

Step 4.1

Raise $10$ to the power of $2$ .

$2 x - 3 = 100 (7 x)$

Step 4.2

Multiply $7$ by $100$ .

$2 x - 3 = 700 x$

Step 5

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

Tap for more steps...

Step 5.1

Subtract $700 x$ from both sides of the equation.

$2 x - 3 - 700 x = 0$

Step 5.2

Subtract $700 x$ from $2 x$ .

$- 698 x - 3 = 0$

Step 6

Add $3$ to both sides of the equation.

$- 698 x = 3$

Step 7

Divide each term in $- 698 x = 3$ by $- 698$ and simplify.

Tap for more steps...

Step 7.1

Divide each term in $- 698 x = 3$ by $- 698$ .

$\frac{- 698 x}{- 698} = \frac{3}{- 698}$

Step 7.2

Simplify the left side.

Tap for more steps...

Step 7.2.1

Cancel the common factor of $- 698$ .

Tap for more steps...

Step 7.2.1.1

Cancel the common factor.

$\frac{- 698 x}{- 698} = \frac{3}{- 698}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{- 698}$

Step 7.3

Simplify the right side.

Tap for more steps...

Step 7.3.1

Move the negative in front of the fraction.

$x = - \frac{3}{698}$

Step 8

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

$No solution$