Trigonometry Examples

$\log_{x} (27) + \log_{x} (3) = 2$

Step 1

Simplify the left side.

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Step 1.1

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

$\log_{x} (27 \cdot 3) = 2$

Step 1.2

Multiply $27$ by $3$ .

$\log_{x} (81) = 2$

Step 2

Rewrite $\log_{x} (81) = 2$ 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 $b^{y} = x$ .

$x^{2} = 81$

Step 3

Solve for $x$ .

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Step 3.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{81}$

Step 3.2

Simplify $\pm \sqrt{81}$ .

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Step 3.2.1

Rewrite $81$ as $9^{2}$ .

$x = \pm \sqrt{9^{2}}$

Step 3.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 9$

Step 3.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 9$

Step 3.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 9$

Step 3.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 9, - 9$

Step 4

Exclude the solutions that do not make $\log_{x} (27) + \log_{x} (3) = 2$ true.

$x = 9$