Calculus Examples

$\log_{3} (1 + x \ln (3))$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$1 + x \ln (3) = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Subtract $1$ from both sides of the equation.

$x \ln (3) = - 1$

Step 1.2.2

Divide each term in $x \ln (3) = - 1$ by $\ln (3)$ and simplify.

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Step 1.2.2.1

Divide each term in $x \ln (3) = - 1$ by $\ln (3)$ .

$\frac{x \ln (3)}{\ln (3)} = \frac{- 1}{\ln (3)}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $\ln (3)$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{x \ln (3)}{\ln (3)} = \frac{- 1}{\ln (3)}$

Step 1.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{\ln (3)}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{\ln (3)}$

Step 1.3

The vertical asymptote occurs at $x = - \frac{1}{\ln (3)}$ .

Vertical Asymptote: $x = - \frac{1}{\ln (3)}$

Step 2

Find the point at $x = 1$ .

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Step 2.1

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

$f (1) = \log_{3} (1 + (1) \ln (3))$

Step 2.2

Simplify the result.

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Step 2.2.1

Multiply $\ln (3)$ by $1$ .

$f (1) = \log_{3} (1 + \ln (3))$

Step 2.2.2

The final answer is $\log_{3} (1 + \ln (3))$ .

$\log_{3} (1 + \ln (3))$

Step 2.3

Convert $\log_{3} (1 + \ln (3))$ to decimal.

$y = 0.67473877$

Step 3

Find the point at $x = 2$ .

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Step 3.1

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

$f (2) = \log_{3} (1 + (2) \ln (3))$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Simplify $(2) \ln (3)$ by moving $2$ inside the logarithm.

$f (2) = \log_{3} (1 + \ln (3^{2}))$

Step 3.2.1.2

Raise $3$ to the power of $2$ .

$f (2) = \log_{3} (1 + \ln (9))$

Step 3.2.2

The final answer is $\log_{3} (1 + \ln (9))$ .

$\log_{3} (1 + \ln (9))$

Step 3.3

Convert $\log_{3} (1 + \ln (9))$ to decimal.

$y = 1.05795568$

Step 4

Find the point at $x = 3$ .

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Step 4.1

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

$f (3) = \log_{3} (1 + (3) \ln (3))$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Simplify $(3) \ln (3)$ by moving $3$ inside the logarithm.

$f (3) = \log_{3} (1 + \ln (3^{3}))$

Step 4.2.1.2

Raise $3$ to the power of $3$ .

$f (3) = \log_{3} (1 + \ln (27))$

Step 4.2.2

The final answer is $\log_{3} (1 + \ln (27))$ .

$\log_{3} (1 + \ln (27))$

Step 4.3

Convert $\log_{3} (1 + \ln (27))$ to decimal.

$y = 1.32680691$

Step 5

The log function can be graphed using the vertical asymptote at $x = - \frac{1}{\ln (3)}$ and the points $(1, 0.67473877), (2, 1.05795568), (3, 1.32680691)$ .

Vertical Asymptote: $x = - \frac{1}{\ln (3)}$

$\begin{array}{c} x & y \\ 1 & 0.675 \\ 2 & 1.058 \\ 3 & 1.327 \end{array}$

Step 6