Solve for x log base 6 of 8x-3 log base 6 of 3=2

$\log_{6} (8 x) - 3 \log_{6} (3) = 2$

Step 1

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

$\log_{6} (8 x) - 3 \log_{6} (3) = 2$

Step 2

Simplify the left side.

Simplify $\log_{6} (8 x) - 3 \log_{6} (3)$ .

Simplify each term.

Simplify $- 3 \log_{6} (3)$ by moving $3$ inside the logarithm.

$\log_{6} (8 x) - \log_{6} (3^{3}) = 2$

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

$\log_{6} (8 x) - \log_{6} (27) = 2$

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

$\log_{6} (\frac{8 x}{27}) = 2$

Step 3

Rewrite $\log_{6} (\frac{8 x}{27}) = 2$ 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 $b^{y} = x$ .

$6^{2} = \frac{8 x}{27}$

Step 4

Solve for $x$ .

Rewrite the equation as $\frac{8 x}{27} = 6^{2}$ .

$\frac{8 x}{27} = 6^{2}$

Multiply both sides of the equation by $\frac{27}{8}$ .

$\frac{27}{8} \cdot \frac{8 x}{27} = \frac{27}{8} \cdot 6^{2}$

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{27}{8} \cdot \frac{8 x}{27}$ .

Cancel the common factor of $27$ .

Cancel the common factor.

$\frac{27}{8} \cdot \frac{8 x}{27} = \frac{27}{8} \cdot 6^{2}$

Rewrite the expression.

$\frac{1}{8} (8 x) = \frac{27}{8} \cdot 6^{2}$

Cancel the common factor of $8$ .

Factor $8$ out of $8 x$ .

$\frac{1}{8} (8 (x)) = \frac{27}{8} \cdot 6^{2}$

Cancel the common factor.

$\frac{1}{8} (8 x) = \frac{27}{8} \cdot 6^{2}$

Rewrite the expression.

$x = \frac{27}{8} \cdot 6^{2}$

Simplify the right side.

Simplify $\frac{27}{8} \cdot 6^{2}$ .

Raise $6$ to the power of $2$ .

$x = \frac{27}{8} \cdot 36$

Cancel the common factor of $4$ .

Factor $4$ out of $8$ .

$x = \frac{27}{4 (2)} \cdot 36$

Factor $4$ out of $36$ .

$x = \frac{27}{4 \cdot 2} \cdot (4 \cdot 9)$

Cancel the common factor.

$x = \frac{27}{4 \cdot 2} \cdot (4 \cdot 9)$

Rewrite the expression.

$x = \frac{27}{2} \cdot 9$

Combine $\frac{27}{2}$ and $9$ .

$x = \frac{27 \cdot 9}{2}$

Multiply $27$ by $9$ .

$x = \frac{243}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{243}{2}$

Decimal Form:

$x = 121.5$

Mixed Number Form:

$x = 121 \frac{1}{2}$

Solve for x log base 6 of 8x-3 log base 6 of 3=2

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