Precalculus Examples

$\log_{5} (x + 3) = 4 + \log_{5} (x - 5)$

Step 1

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

$\log_{5} (x + 3) - \log_{5} (x - 5) = 4$

Step 2

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

$\log_{5} (\frac{x + 3}{x - 5}) = 4$

Step 3

Rewrite $\log_{5} (\frac{x + 3}{x - 5}) = 4$ 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 $b^{y} = x$ .

$5^{4} = \frac{x + 3}{x - 5}$

Step 4

Cross multiply to remove the fraction.

$x + 3 = 5^{4} (x - 5)$

Step 5

Simplify $5^{4} (x - 5)$ .

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Step 5.1

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

$x + 3 = 625 (x - 5)$

Step 5.2

Apply the distributive property.

$x + 3 = 625 x + 625 \cdot - 5$

Step 5.3

Multiply $625$ by $- 5$ .

$x + 3 = 625 x - 3125$

Step 6

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

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Step 6.1

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

$x + 3 - 625 x = - 3125$

Step 6.2

Subtract $625 x$ from $x$ .

$- 624 x + 3 = - 3125$

Step 7

Factor $3$ out of $- 624 x + 3$ .

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Step 7.1

Factor $3$ out of $- 624 x$ .

$3 (- 208 x) + 3 = - 3125$

Step 7.2

Factor $3$ out of $3$ .

$3 (- 208 x) + 3 (1) = - 3125$

Step 7.3

Factor $3$ out of $3 (- 208 x) + 3 (1)$ .

$3 (- 208 x + 1) = - 3125$

Step 8

Divide each term in $3 (- 208 x + 1) = - 3125$ by $3$ and simplify.

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Step 8.1

Divide each term in $3 (- 208 x + 1) = - 3125$ by $3$ .

$\frac{3 (- 208 x + 1)}{3} = \frac{- 3125}{3}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $3$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{3 (- 208 x + 1)}{3} = \frac{- 3125}{3}$

Step 8.2.1.2

Divide $- 208 x + 1$ by $1$ .

$- 208 x + 1 = \frac{- 3125}{3}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Move the negative in front of the fraction.

$- 208 x + 1 = - \frac{3125}{3}$

Step 9

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

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Step 9.1

Subtract $1$ from both sides of the equation.

$- 208 x = - \frac{3125}{3} - 1$

Step 9.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- 208 x = - \frac{3125}{3} - 1 \cdot \frac{3}{3}$

Step 9.3

Combine $- 1$ and $\frac{3}{3}$ .

$- 208 x = - \frac{3125}{3} + \frac{- 1 \cdot 3}{3}$

Step 9.4

Combine the numerators over the common denominator.

$- 208 x = \frac{- 3125 - 1 \cdot 3}{3}$

Step 9.5

Simplify the numerator.

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Step 9.5.1

Multiply $- 1$ by $3$ .

$- 208 x = \frac{- 3125 - 3}{3}$

Step 9.5.2

Subtract $3$ from $- 3125$ .

$- 208 x = \frac{- 3128}{3}$

Step 9.6

Move the negative in front of the fraction.

$- 208 x = - \frac{3128}{3}$

Step 10

Divide each term in $- 208 x = - \frac{3128}{3}$ by $- 208$ and simplify.

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Step 10.1

Divide each term in $- 208 x = - \frac{3128}{3}$ by $- 208$ .

$\frac{- 208 x}{- 208} = \frac{- \frac{3128}{3}}{- 208}$

Step 10.2

Simplify the left side.

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Step 10.2.1

Cancel the common factor of $- 208$ .

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Step 10.2.1.1

Cancel the common factor.

$\frac{- 208 x}{- 208} = \frac{- \frac{3128}{3}}{- 208}$

Step 10.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{3128}{3}}{- 208}$

Step 10.3

Simplify the right side.

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Step 10.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{3128}{3} \cdot \frac{1}{- 208}$

Step 10.3.2

Cancel the common factor of $8$ .

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Step 10.3.2.1

Move the leading negative in $- \frac{3128}{3}$ into the numerator.

$x = \frac{- 3128}{3} \cdot \frac{1}{- 208}$

Step 10.3.2.2

Factor $8$ out of $- 3128$ .

$x = \frac{8 (- 391)}{3} \cdot \frac{1}{- 208}$

Step 10.3.2.3

Factor $8$ out of $- 208$ .

$x = \frac{8 \cdot - 391}{3} \cdot \frac{1}{8 \cdot - 26}$

Step 10.3.2.4

Cancel the common factor.

$x = \frac{8 \cdot - 391}{3} \cdot \frac{1}{8 \cdot - 26}$

Step 10.3.2.5

Rewrite the expression.

$x = \frac{- 391}{3} \cdot \frac{1}{- 26}$

Step 10.3.3

Multiply $\frac{- 391}{3}$ by $\frac{1}{- 26}$ .

$x = \frac{- 391}{3 \cdot - 26}$

Step 10.3.4

Multiply $3$ by $- 26$ .

$x = \frac{- 391}{- 78}$

Step 10.3.5

Dividing two negative values results in a positive value.

$x = \frac{391}{78}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$x = \frac{391}{78}$

Decimal Form:

$x = 5.01282051 \dots$

Mixed Number Form:

$x = 5 \frac{1}{78}$