Precalculus Examples

$\log_{3} (2 y) = 4 - \log_{3} (x)$

Step 1

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

$\log_{3} (2 y) + \log_{3} (x) = 4$

Step 2

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

$\log_{3} (2 y x) = 4$

Step 3

Rewrite $\log_{3} (2 y x) = 4$ 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 $b^{y} = x$ .

$3^{4} = 2 y x$

Step 4

Solve for $y$ .

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Step 4.1

Rewrite the equation as $2 y x = 3^{4}$ .

$2 y x = 3^{4}$

Step 4.2

Divide each term in $2 y x = 3^{4}$ by $2 x$ and simplify.

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Step 4.2.1

Divide each term in $2 y x = 3^{4}$ by $2 x$ .

$\frac{2 y x}{2 x} = \frac{3^{4}}{2 x}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $2$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{2 y x}{2 x} = \frac{3^{4}}{2 x}$

Step 4.2.2.1.2

Rewrite the expression.

$\frac{y x}{x} = \frac{3^{4}}{2 x}$

Step 4.2.2.2

Cancel the common factor of $x$ .

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Step 4.2.2.2.1

Cancel the common factor.

$\frac{y x}{x} = \frac{3^{4}}{2 x}$

Step 4.2.2.2.2

Divide $y$ by $1$ .

$y = \frac{3^{4}}{2 x}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Raise $3$ to the power of $4$ .

$y = \frac{81}{2 x}$

Step 5

To rewrite as a function of $x$ , write the equation so that $y$ is by itself on one side of the equal sign and an expression involving only $x$ is on the other side.

$f (x) = \frac{81}{2 x}$