Calculus Examples

$\log (9 x)$

Find the asymptotes.

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Set the argument of the logarithm equal to zero.

$9 x = 0$

Divide each term in $9 x = 0$ by $9$ and simplify.

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Divide each term in $9 x = 0$ by $9$ .

$\frac{9 x}{9} = \frac{0}{9}$

Simplify the left side.

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Cancel the common factor of $9$ .

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Cancel the common factor.

$\frac{9 x}{9} = \frac{0}{9}$

Divide $x$ by $1$ .

$x = \frac{0}{9}$

Simplify the right side.

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Divide $0$ by $9$ .

$x = 0$

The vertical asymptote occurs at $x = 0$ .

Vertical Asymptote: $x = 0$

Find the point at $x = 1$ .

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

$f (1) = \log (9 (1))$

Simplify the result.

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Multiply $9$ by $1$ .

$f (1) = \log (9)$

The final answer is $\log (9)$ .

$\log (9)$

Convert $\log (9)$ to decimal.

$y = 0.9542425$

Find the point at $x = 2$ .

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

$f (2) = \log (9 (2))$

Simplify the result.

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Multiply $9$ by $2$ .

$f (2) = \log (18)$

The final answer is $\log (18)$ .

$\log (18)$

Convert $\log (18)$ to decimal.

$y = 1.2552725$

Find the point at $x = 3$ .

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Replace the variable $x$ with $3$ in the expression.

$f (3) = \log (9 (3))$

Simplify the result.

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Multiply $9$ by $3$ .

$f (3) = \log (27)$

The final answer is $\log (27)$ .

$\log (27)$

Convert $\log (27)$ to decimal.

$y = 1.43136376$

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(1, 0.9542425), (2, 1.2552725), (3, 1.43136376)$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & 0.954 \\ 2 & 1.255 \\ 3 & 1.431 \end{array}$