Solve for x log base 6 of 2x-5+1 = log base 6 of 7x+10

$\log_{6} (2 x - 5) + 1 = \log_{6} (7 x + 10)$

Step 1

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

$\log_{6} (2 x - 5) - \log_{6} (7 x + 10) = - 1$

Step 2

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

$\log_{6} (\frac{2 x - 5}{7 x + 10}) = - 1$

Step 3

Rewrite $\log_{6} (\frac{2 x - 5}{7 x + 10}) = - 1$ 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 $b^{y} = x$ .

$6^{- 1} = \frac{2 x - 5}{7 x + 10}$

Step 4

Cross multiply to remove the fraction.

$2 x - 5 = 6^{- 1} (7 x + 10)$

Step 5

Simplify $6^{- 1} (7 x + 10)$ .

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$2 x - 5 = \frac{1}{6} (7 x + 10)$

Apply the distributive property.

$2 x - 5 = \frac{1}{6} (7 x) + \frac{1}{6} \cdot 10$

Multiply $\frac{1}{6} (7 x)$ .

Combine $7$ and $\frac{1}{6}$ .

$2 x - 5 = \frac{7}{6} x + \frac{1}{6} \cdot 10$

Combine $\frac{7}{6}$ and $x$ .

$2 x - 5 = \frac{7 x}{6} + \frac{1}{6} \cdot 10$

Cancel the common factor of $2$ .

Factor $2$ out of $6$ .

$2 x - 5 = \frac{7 x}{6} + \frac{1}{2 (3)} \cdot 10$

Factor $2$ out of $10$ .

$2 x - 5 = \frac{7 x}{6} + \frac{1}{2 \cdot 3} \cdot (2 \cdot 5)$

Cancel the common factor.

$2 x - 5 = \frac{7 x}{6} + \frac{1}{2 \cdot 3} \cdot (2 \cdot 5)$

Rewrite the expression.

$2 x - 5 = \frac{7 x}{6} + \frac{1}{3} \cdot 5$

Combine $\frac{1}{3}$ and $5$ .

$2 x - 5 = \frac{7 x}{6} + \frac{5}{3}$

Step 6

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

Subtract $\frac{7 x}{6}$ from both sides of the equation.

$2 x - 5 - \frac{7 x}{6} = \frac{5}{3}$

To write $2 x$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$2 x \cdot \frac{6}{6} - \frac{7 x}{6} - 5 = \frac{5}{3}$

Combine $2 x$ and $\frac{6}{6}$ .

$\frac{2 x \cdot 6}{6} - \frac{7 x}{6} - 5 = \frac{5}{3}$

Combine the numerators over the common denominator.

$\frac{2 x \cdot 6 - 7 x}{6} - 5 = \frac{5}{3}$

Simplify each term.

Simplify the numerator.

Factor $x$ out of $2 x \cdot 6 - 7 x$ .

Factor $x$ out of $2 x \cdot 6$ .

$\frac{x (2 \cdot 6) - 7 x}{6} - 5 = \frac{5}{3}$

Factor $x$ out of $- 7 x$ .

$\frac{x (2 \cdot 6) + x \cdot - 7}{6} - 5 = \frac{5}{3}$

Factor $x$ out of $x (2 \cdot 6) + x \cdot - 7$ .

$\frac{x (2 \cdot 6 - 7)}{6} - 5 = \frac{5}{3}$

Multiply $2$ by $6$ .

$\frac{x (12 - 7)}{6} - 5 = \frac{5}{3}$

Subtract $7$ from $12$ .

$\frac{x \cdot 5}{6} - 5 = \frac{5}{3}$

Move $5$ to the left of $x$ .

$\frac{5 x}{6} - 5 = \frac{5}{3}$

Step 7

Factor $5$ out of $\frac{5 x}{6} - 5$ .

Factor $5$ out of $\frac{5 x}{6}$ .

$5 (\frac{x}{6}) - 5 = \frac{5}{3}$

Factor $5$ out of $- 5$ .

$5 (\frac{x}{6}) + 5 (- 1) = \frac{5}{3}$

Factor $5$ out of $5 \frac{x}{6} + 5 (- 1)$ .

$5 (\frac{x}{6} - 1) = \frac{5}{3}$

Step 8

Divide each term in $5 (\frac{x}{6} - 1) = \frac{5}{3}$ by $5$ and simplify.

Divide each term in $5 (\frac{x}{6} - 1) = \frac{5}{3}$ by $5$ .

$\frac{5 (\frac{x}{6} - 1)}{5} = \frac{\frac{5}{3}}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 (\frac{x}{6} - 1)}{5} = \frac{\frac{5}{3}}{5}$

Divide $\frac{x}{6} - 1$ by $1$ .

$\frac{x}{6} - 1 = \frac{\frac{5}{3}}{5}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$\frac{x}{6} - 1 = \frac{5}{3} \cdot \frac{1}{5}$

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{x}{6} - 1 = \frac{5}{3} \cdot \frac{1}{5}$

Rewrite the expression.

$\frac{x}{6} - 1 = \frac{1}{3}$

Step 9

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

Add $1$ to both sides of the equation.

$\frac{x}{6} = \frac{1}{3} + 1$

Write $1$ as a fraction with a common denominator.

$\frac{x}{6} = \frac{1}{3} + \frac{3}{3}$

Combine the numerators over the common denominator.

$\frac{x}{6} = \frac{1 + 3}{3}$

Add $1$ and $3$ .

$\frac{x}{6} = \frac{4}{3}$

Step 10

Multiply both sides of the equation by $6$ .

$6 \frac{x}{6} = 6 (\frac{4}{3})$

Step 11

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$6 \frac{x}{6} = 6 (\frac{4}{3})$

Rewrite the expression.

$x = 6 (\frac{4}{3})$

Simplify the right side.

Simplify $6 (\frac{4}{3})$ .

Cancel the common factor of $3$ .

Factor $3$ out of $6$ .

$x = 3 (2) \frac{4}{3}$

Cancel the common factor.

$x = 3 \cdot 2 \frac{4}{3}$

Rewrite the expression.

$x = 2 \cdot 4$

Multiply $2$ by $4$ .

$x = 8$

Solve for x log base 6 of 2x-5+1 = log base 6 of 7x+10

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