Precalculus Examples

$\log (\sqrt{\frac{10}{1000}})$

Step 1

Cancel the common factor of $10$ and $1000$ .

Tap for more steps...

Step 1.1

Factor $10$ out of $10$ .

$\log (\sqrt{\frac{10 (1)}{1000}})$

Step 1.2

Cancel the common factors.

Tap for more steps...

Step 1.2.1

Factor $10$ out of $1000$ .

$\log (\sqrt{\frac{10 \cdot 1}{10 \cdot 100}})$

Step 1.2.2

Cancel the common factor.

$\log (\sqrt{\frac{10 \cdot 1}{10 \cdot 100}})$

Step 1.2.3

Rewrite the expression.

$\log (\sqrt{\frac{1}{100}})$

Step 2

Rewrite $\sqrt{\frac{1}{100}}$ as $\frac{\sqrt{1}}{\sqrt{100}}$ .

$\log (\frac{\sqrt{1}}{\sqrt{100}})$

Step 3

Any root of $1$ is $1$ .

$\log (\frac{1}{\sqrt{100}})$

Step 4

Simplify the denominator.

Tap for more steps...

Step 4.1

Rewrite $100$ as $10^{2}$ .

$\log (\frac{1}{\sqrt{10^{2}}})$

Step 4.2

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

$\log (\frac{1}{10})$

Step 5

Logarithm base $10$ of $\frac{1}{10}$ is $- 1$ .

Tap for more steps...

Step 5.1

Rewrite as an equation.

$\log (\frac{1}{10}) = x$

Step 5.2

Rewrite $\log (\frac{1}{10}) = x$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b$ does not equal $1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$10^{x} = \frac{1}{10}$

Step 5.3

Create equivalent expressions in the equation that all have equal bases.

$10^{x} = 10^{- 1}$

Step 5.4

Since the bases are the same, the two expressions are only equal if the exponents are also equal.

$x = - 1$

Step 5.5

The variable $x$ is equal to $- 1$ .

$- 1$