Pre-Algebra Examples

$\log (\log (100000^{2 x}))$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$\log (100000^{2 x}) = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Expand $\log (100000^{2 x})$ .

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Step 1.2.1.1

Expand $\log (100000^{2 x})$ by moving $2 x$ outside the logarithm.

$2 x \log (100000)$

Step 1.2.1.2

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

$2 x \cdot 5$

Step 1.2.1.3

Multiply $5$ by $2$ .

$10 x$

Step 1.2.2

The expanded equation is $10 x = 0$ .

$10 x = 0$

Step 1.2.3

Divide each term in $10 x = 0$ by $10$ and simplify.

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Step 1.2.3.1

Divide each term in $10 x = 0$ by $10$ .

$\frac{10 x}{10} = \frac{0}{10}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $10$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{10 x}{10} = \frac{0}{10}$

Step 1.2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{10}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $0$ by $10$ .

$x = 0$

Step 1.3

The vertical asymptote occurs at $x = 0$ .

Vertical Asymptote: $x = 0$

Step 2

Find the point at $x = 1$ .

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Step 2.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = \log (\log (100000^{2 (1)}))$

Step 2.2

Simplify the result.

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Step 2.2.1

Multiply $2$ by $1$ .

$f (1) = \log (\log (100000^{2}))$

Step 2.2.2

Raise $100000$ to the power of $2$ .

$f (1) = \log (\log (10000000000))$

Step 2.2.3

Logarithm base $10$ of $10000000000$ is $10$ .

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Step 2.2.3.1

Rewrite as an equation.

$\log (\log (10000000000) = x)$

Step 2.2.3.2

Rewrite $\log (10000000000) = x$ 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 $b^{y} = x$ .

$\log (10^{x} = 10000000000)$

Step 2.2.3.3

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

$\log (10^{x} = 10^{10})$

Step 2.2.3.4

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

$\log (x = 10)$

Step 2.2.3.5

The variable $x$ is equal to $10$ .

$f (1) = \log (10)$

Step 2.2.4

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

$f (1) = 1$

Step 2.2.5

The final answer is $1$ .

$1$

Step 2.3

Convert $1$ to decimal.

$y = 1$

Step 3

Find the point at $x = 10$ .

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Step 3.1

Replace the variable $x$ with $10$ in the expression.

$f (10) = \log (\log (100000^{2 (10)}))$

Step 3.2

Simplify the result.

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Step 3.2.1

Multiply $2$ by $10$ .

$f (10) = \log (\log (100000^{20}))$

Step 3.2.2

The final answer is $\log (\log (100000^{20}))$ .

$\log (\log (100000^{20}))$

Step 3.3

Convert $\log (\log (100000^{20}))$ to decimal.

$y = 2$

Step 4

Find the point at $x = 2$ .

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Step 4.1

Replace the variable $x$ with $2$ in the expression.

$f (2) = \log (\log (100000^{2 (2)}))$

Step 4.2

Simplify the result.

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Step 4.2.1

Multiply $2$ by $2$ .

$f (2) = \log (\log (100000^{4}))$

Step 4.2.2

Raise $100000$ to the power of $4$ .

$f (2) = \log (\log (100000000000000000000))$

Step 4.2.3

Logarithm base $10$ of $100000000000000000000$ is $20$ .

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Step 4.2.3.1

Rewrite as an equation.

$\log (\log (100000000000000000000) = x)$

Step 4.2.3.2

Rewrite $\log (100000000000000000000) = x$ 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 $b^{y} = x$ .

$\log (10^{x} = 100000000000000000000)$

Step 4.2.3.3

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

$\log (10^{x} = 10^{20})$

Step 4.2.3.4

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

$\log (x = 20)$

Step 4.2.3.5

The variable $x$ is equal to $20$ .

$f (2) = \log (20)$

Step 4.2.4

The final answer is $\log (20)$ .

$\log (20)$

Step 4.3

Convert $\log (20)$ to decimal.

$y = 1.30102999$

Step 5

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(1, 1), (10, 2), (2, 1.30102999)$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & 1 \\ 2 & 1.301 \\ 10 & 2 \end{array}$

Step 6