Precalculus Examples

$\frac{\log_{5} (2500)}{2}$

Step 1

Rewrite $\frac{\log_{5} (2500)}{2}$ as $\frac{1}{2} \log_{5} (2500)$ .

$\frac{1}{2} \log_{5} (2500)$

Step 2

Simplify $\frac{1}{2} \log_{5} (2500)$ by moving $\frac{1}{2}$ inside the logarithm.

$\log_{5} (2500^{\frac{1}{2}})$

Step 3

Simplify the expression.

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Step 3.1

Rewrite $2500$ as $50^{2}$ .

$\log_{5} ({(50^{2})}^{\frac{1}{2}})$

Step 3.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log_{5} (50^{2 (\frac{1}{2})})$

Step 4

Cancel the common factor of $2$ .

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Step 4.1

Cancel the common factor.

$\log_{5} (50^{2 (\frac{1}{2})})$

Step 4.2

Rewrite the expression.

$\log_{5} (50^{1})$

Step 5

Evaluate the exponent.

$\log_{5} (50)$

Step 6

Rewrite $\log_{5} (50)$ using the change of base formula.

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Step 6.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 6.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\log (50)}{\log (5)}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (50)}{\log (5)}$

Decimal Form:

$2.43067655 \dots$