Precalculus Examples

$\log (100^{\frac{5}{2}})$

Step 1

Simplify the expression.

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Step 1.1

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

$\log ({(10^{2})}^{\frac{5}{2}})$

Step 1.2

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

$\log (10^{2 (\frac{5}{2})})$

Step 2

Cancel the common factor of $2$ .

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Step 2.1

Cancel the common factor.

$\log (10^{2 (\frac{5}{2})})$

Step 2.2

Rewrite the expression.

$\log (10^{5})$

Step 3

Raise $10$ to the power of $5$ .

$\log (100000)$

Step 4

Logarithm base $10$ of $100000$ is $5$ .

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Step 4.1

Rewrite as an equation.

$\log (100000) = x$

Step 4.2

Rewrite $\log (100000) = x$ 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 $b^{y} = x$ .

$10^{x} = 100000$

Step 4.3

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

$10^{x} = 10^{5}$

Step 4.4

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

$x = 5$

Step 4.5

The variable $x$ is equal to $5$ .

$5$