Precalculus Examples

$\log (\log (\log (10^{10^{100}})))$

Step 1

Use logarithm rules to move $10^{100}$ out of the exponent.

$\log (\log (10^{100} \log (10)))$

Step 2

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

$\log (\log (10^{100} \cdot 1))$

Step 3

Multiply $10^{100}$ by $1$ .

$\log (\log (10^{100}))$

Step 4

Use logarithm rules to move $100$ out of the exponent.

$\log (100 \log (10))$

Step 5

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

$\log (100 \cdot 1)$

Step 6

Multiply $100$ by $1$ .

$\log (100)$

Step 7

Logarithm base $10$ of $100$ is $2$ .

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Step 7.1

Rewrite as an equation.

$\log (100) = x$

Step 7.2

Rewrite $\log (100) = x$ 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 $b^{y} = x$ .

$10^{x} = 100$

Step 7.3

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

$10^{x} = 10^{2}$

Step 7.4

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

$x = 2$

Step 7.5

The variable $x$ is equal to $2$ .

$2$