Precalculus Examples

$\log (\sqrt{1000000})$

Step 1

Rewrite $1000000$ as $1000^{2}$ .

$\log (\sqrt{1000^{2}})$

Step 2

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

$\log (1000)$

Step 3

Logarithm base $10$ of $1000$ is $3$ .

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Step 3.1

Rewrite as an equation.

$\log (1000) = x$

Step 3.2

Rewrite $\log (1000) = x$ 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 $b^{y} = x$ .

$10^{x} = 1000$

Step 3.3

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

$10^{x} = 10^{3}$

Step 3.4

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

$x = 3$

Step 3.5

The variable $x$ is equal to $3$ .

$3$