Calculus Examples

$\log_{5} (\frac{125 \cdot \sqrt{625}}{5^{2} \cdot 25^{2}})$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $625$ as $25^{2}$ .

$\log_{5} (\frac{125 \sqrt{25^{2}}}{5^{2} \cdot 25^{2}})$

Step 1.2

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

$\log_{5} (\frac{125 \cdot 25}{5^{2} \cdot 25^{2}})$

Step 2

Simplify the denominator.

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Step 2.1

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

$\log_{5} (\frac{125 \cdot 25}{5^{2} \cdot {(5^{2})}^{2}})$

Step 2.2

Multiply the exponents in ${(5^{2})}^{2}$ .

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Step 2.2.1

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

$\log_{5} (\frac{125 \cdot 25}{5^{2} \cdot 5^{2 \cdot 2}})$

Step 2.2.2

Multiply $2$ by $2$ .

$\log_{5} (\frac{125 \cdot 25}{5^{2} \cdot 5^{4}})$

Step 2.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\log_{5} (\frac{125 \cdot 25}{5^{2 + 4}})$

Step 2.4

Add $2$ and $4$ .

$\log_{5} (\frac{125 \cdot 25}{5^{6}})$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Multiply $125$ by $25$ .

$\log_{5} (\frac{3125}{5^{6}})$

Step 3.2

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

$\log_{5} (\frac{3125}{15625})$

Step 3.3

Reduce the expression $\frac{3125}{15625}$ by cancelling the common factors.

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Step 3.3.1

Factor $3125$ out of $3125$ .

$\log_{5} (\frac{3125 (1)}{15625})$

Step 3.3.2

Factor $3125$ out of $15625$ .

$\log_{5} (\frac{3125 \cdot 1}{3125 \cdot 5})$

Step 3.3.3

Cancel the common factor.

$\log_{5} (\frac{3125 \cdot 1}{3125 \cdot 5})$

Step 3.3.4

Rewrite the expression.

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

Step 4

Rewrite $\frac{1}{5}$ as $1 \cdot 5^{- 1}$ .

$\log_{5} (1 \cdot 5^{- 1})$

Step 5

Rewrite $\log_{5} (1 \cdot 5^{- 1})$ as $\log_{5} (1) + \log_{5} (5^{- 1})$ .

$\log_{5} (1) + \log_{5} (5^{- 1})$

Step 6

Use logarithm rules to move $- 1$ out of the exponent.

$\log_{5} (1) - \log_{5} (5)$

Step 7

Logarithm base $5$ of $5$ is $1$ .

$\log_{5} (1) - 1 \cdot 1$

Step 8

Multiply $- 1$ by $1$ .

$\log_{5} (1) - 1$

Step 9

Logarithm base $5$ of $1$ is $0$ .

$0 - 1$

Step 10

Subtract $1$ from $0$ .

$- 1$