Precalculus Examples

$\log (\log (10^{100000}))$

Step 1

Use logarithm rules to move $100000$ out of the exponent.

$\log (100000 \log (10))$

Step 2

Logarithm base $10$ of $10$ is $1$ .

$\log (100000 \cdot 1)$

Step 3

Multiply $100000$ by $1$ .

$\log (100000)$

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 $5$ .

$5$