Algebra Examples

$\log_{3} (x + 1) - \log_{3} (x) = 3$

Step 1

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

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

Step 2

Rewrite $\log_{3} (\frac{x + 1}{x}) = 3$ 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 $b^{y} = x$ .

$3^{3} = \frac{x + 1}{x}$

Step 3

Cross multiply to remove the fraction.

$x + 1 = 3^{3} (x)$

Step 4

Raise $3$ to the power of $3$ .

$x + 1 = 27 x$

Step 5

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

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Step 5.1

Subtract $27 x$ from both sides of the equation.

$x + 1 - 27 x = 0$

Step 5.2

Subtract $27 x$ from $x$ .

$- 26 x + 1 = 0$

Step 6

Subtract $1$ from both sides of the equation.

$- 26 x = - 1$

Step 7

Divide each term in $- 26 x = - 1$ by $- 26$ and simplify.

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Step 7.1

Divide each term in $- 26 x = - 1$ by $- 26$ .

$\frac{- 26 x}{- 26} = \frac{- 1}{- 26}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $- 26$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{- 26 x}{- 26} = \frac{- 1}{- 26}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{- 26}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Dividing two negative values results in a positive value.

$x = \frac{1}{26}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{26}$

Decimal Form:

$x = 0.0\overline{384615}$