Solve for x log base 5 of x+1- log base 5 of x=4

$\log_{5} (x + 1) - \log_{5} (x) = 4$

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

$\log_{5} (\frac{x + 1}{x}) = 4$

Rewrite $\log_{5} (\frac{x + 1}{x}) = 4$ 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 $b^{y} = x$ .

$5^{4} = \frac{x + 1}{x}$

Cross multiply to remove the fraction.

$x + 1 = 5^{4} (x)$

Raise $5$ to the power of $4$ .

$x + 1 = 625 x$

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

Subtract $625 x$ from both sides of the equation.

$x + 1 - 625 x = 0$

Subtract $625 x$ from $x$ .

$- 624 x + 1 = 0$

Subtract $1$ from both sides of the equation.

$- 624 x = - 1$

Divide each term in $- 624 x = - 1$ by $- 624$ and simplify.

Divide each term in $- 624 x = - 1$ by $- 624$ .

$\frac{- 624 x}{- 624} = \frac{- 1}{- 624}$

Simplify the left side.

Cancel the common factor of $- 624$ .

Cancel the common factor.

$\frac{- 624 x}{- 624} = \frac{- 1}{- 624}$

Divide $x$ by $1$ .

$x = \frac{- 1}{- 624}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x = \frac{1}{624}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{624}$

Decimal Form:

$x = 0.00160256 \dots$

Solve for x log base 5 of x+1- log base 5 of x=4

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