Trigonometry Examples

$\log_{b} (25) = 2$

Step 1

Rewrite $\log_{b} (25) = 2$ 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 $b^{y} = x$ .

$b^{2} = 25$

Step 2

Solve for $b$ .

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Step 2.1

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

$b = \pm \sqrt{25}$

Step 2.2

Simplify $\pm \sqrt{25}$ .

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Step 2.2.1

Rewrite $25$ as $5^{2}$ .

$b = \pm \sqrt{5^{2}}$

Step 2.2.2

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

$b = \pm 5$

Step 2.3

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

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Step 2.3.1

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

$b = 5$

Step 2.3.2

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

$b = - 5$

Step 2.3.3

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

$b = 5, - 5$

Step 3

Exclude the solutions that do not make $\log_{b} (25) = 2$ true.

$b = 5$