Algebra Examples

$\log_{6} (4 x) + 2 \log_{6} (9) = 5$

Step 1

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

$\log_{6} (4 x) + 2 \log_{6} (9) = 5$

Step 2

Simplify the left side.

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Step 2.1

Simplify $\log_{6} (4 x) + 2 \log_{6} (9)$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Simplify $2 \log_{6} (9)$ by moving $2$ inside the logarithm.

$\log_{6} (4 x) + \log_{6} (9^{2}) = 5$

Step 2.1.1.2

Raise $9$ to the power of $2$ .

$\log_{6} (4 x) + \log_{6} (81) = 5$

Step 2.1.2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log_{6} (4 x \cdot 81) = 5$

Step 2.1.3

Multiply $81$ by $4$ .

$\log_{6} (324 x) = 5$

Step 3

Rewrite $\log_{6} (324 x) = 5$ 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 $b^{y} = x$ .

$6^{5} = 324 x$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $324 x = 6^{5}$ .

$324 x = 6^{5}$

Step 4.2

Divide each term in $324 x = 6^{5}$ by $324$ and simplify.

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Step 4.2.1

Divide each term in $324 x = 6^{5}$ by $324$ .

$\frac{324 x}{324} = \frac{6^{5}}{324}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $324$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{324 x}{324} = \frac{6^{5}}{324}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{6^{5}}{324}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Raise $6$ to the power of $5$ .

$x = \frac{7776}{324}$

Step 4.2.3.2

Divide $7776$ by $324$ .

$x = 24$