Trigonometry Examples

$5 f (x) = 6 + 3 \log (5)$

Step 1

Find the asymptotes.

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Step 1.1

Divide each term in $5 y x = 6 + \log (125)$ by $5 x$ and simplify.

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Step 1.1.1

Divide each term in $5 y x = 6 + \log (125)$ by $5 x$ .

$\frac{5 y x}{5 x} = \frac{6}{5 x} + \frac{\log (125)}{5 x}$

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

Cancel the common factor of $5$ .

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Step 1.1.2.1.1

Cancel the common factor.

$\frac{5 y x}{5 x} = \frac{6}{5 x} + \frac{\log (125)}{5 x}$

Step 1.1.2.1.2

Rewrite the expression.

$\frac{y x}{x} = \frac{6}{5 x} + \frac{\log (125)}{5 x}$

Step 1.1.2.2

Cancel the common factor of $x$ .

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Step 1.1.2.2.1

Cancel the common factor.

$\frac{y x}{x} = \frac{6}{5 x} + \frac{\log (125)}{5 x}$

Step 1.1.2.2.2

Divide $y$ by $1$ .

$y = \frac{6}{5 x} + \frac{\log (125)}{5 x}$

Step 1.2

Find where the expression $\frac{6}{5 x} + \frac{\log (125)}{5 x}$ is undefined.

$x = 0$

Step 1.3

Evaluate $lim_{x \to \infty} \frac{6 + \log (125)}{5 x}$ to find the horizontal asymptote.

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Step 1.3.1

Move the term $\frac{6 + \log (125)}{5}$ outside of the limit because it is constant with respect to $x$ .

$\frac{6 + \log (125)}{5} lim_{x \to \infty} \frac{1}{x}$

Step 1.3.2

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x}$ approaches $0$ .

$\frac{6 + \log (125)}{5} \cdot 0$

Step 1.3.3

Multiply $\frac{6 + \log (125)}{5}$ by $0$ .

$0$

Step 1.4

List the horizontal asymptotes:

$y = 0$

Step 1.5

No oblique asymptotes are present for logarithmic and trigonometric functions.

No Oblique Asymptotes

Step 1.6

This is the set of all asymptotes.

Vertical Asymptotes: $x = 0$

Horizontal Asymptotes: $y = 0$

Vertical Asymptotes: $x = 0$

Horizontal Asymptotes: $y = 0$

Step 2

Find the point at $x = 1$ .

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Step 2.1

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

$f (1) = 6 + 3 \log (5)$

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Simplify $3 \log (5)$ by moving $3$ inside the logarithm.

$f (1) = 6 + \log (5^{3})$

Step 2.2.1.2

Raise $5$ to the power of $3$ .

$f (1) = 6 + \log (125)$

Step 2.2.2

The final answer is $6 + \log (125)$ .

$6 + \log (125)$

Step 2.3

Convert $6 + \log (125)$ to decimal.

$y = 8.09691001$

Step 3

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(1, 8.09691001), (2, 8.09691001), (3, 8.09691001)$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & 8.097 \\ 2 & 8.097 \\ 3 & 8.097 \end{array}$

Step 4