Find where the expression <math><mstyle displaystyle="true"><msub><mi>log</mi><mrow><mn>5</mn></mrow></msub><mrow><mo>(</mo><msup><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mn>3</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow><mo>-</mo><mn>1</mn></mstyle></math> is undefined.

Ignoring the logarithm, consider the rational function <math><mstyle displaystyle="true"><mi>R</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mi>a</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow><mrow><mi>b</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>m</mi></mrow></msup></mrow></mfrac></mstyle></math> where <math><mstyle displaystyle="true"><mi>n</mi></mstyle></math> is the degree of the numerator and <math><mstyle displaystyle="true"><mi>m</mi></mstyle></math> is the degree of the denominator.

1. If <math><mstyle displaystyle="true"><mi>n</mi><mo><</mo><mi>m</mi></mstyle></math> , then the x-axis, <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mn>0</mn></mstyle></math> , is the horizontal asymptote.

2. If <math><mstyle displaystyle="true"><mi>n</mi><mo>=</mo><mi>m</mi></mstyle></math> , then the horizontal asymptote is the line <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mfrac><mrow><mi>a</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math> .

3. If <math><mstyle displaystyle="true"><mi>n</mi><mo>></mo><mi>m</mi></mstyle></math> , then there is no horizontal asymptote (there is an oblique asymptote).

There are no horizontal asymptotes because <math><mstyle displaystyle="true"><mi>Q</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> is <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

No Horizontal Asymptotes

No oblique asymptotes are present for logarithmic and trigonometric functions.

No Oblique Asymptotes

This is the set of all asymptotes.

Vertical Asymptotes: <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mo>-</mo><mn>3</mn></mstyle></math>

No Horizontal Asymptotes

Vertical Asymptotes: <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mo>-</mo><mn>3</mn></mstyle></math>

No Horizontal Asymptotes

Replace the variable <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> with <math><mstyle displaystyle="true"><mo>-</mo><mn>2</mn></mstyle></math> in the expression.

Simplify the result.

Simplify each term.

Add <math><mstyle displaystyle="true"><mo>-</mo><mn>2</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

Logarithm base <math><mstyle displaystyle="true"><mn>5</mn></mstyle></math> of <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> is <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> from <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

The final answer is <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> .

Convert <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> to decimal.

Replace the variable <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> with <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> in the expression.

Simplify the result.

Simplify each term.

Add <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

Logarithm base <math><mstyle displaystyle="true"><mn>5</mn></mstyle></math> of <math><mstyle displaystyle="true"><mn>5</mn></mstyle></math> is <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> from <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

The final answer is <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Convert <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> to decimal.

Replace the variable <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> with <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> in the expression.

Simplify the result.

Simplify each term.

Add <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

Simplify <math><mstyle displaystyle="true"><mn>2</mn><msub><mi>log</mi><mrow><mn>5</mn></mrow></msub><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mstyle></math> by moving <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> inside the logarithm.

Raise <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

The final answer is <math><mstyle displaystyle="true"><msub><mi>log</mi><mrow><mn>5</mn></mrow></msub><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow><mo>-</mo><mn>1</mn></mstyle></math> .

Convert <math><mstyle displaystyle="true"><msub><mi>log</mi><mrow><mn>5</mn></mrow></msub><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow><mo>-</mo><mn>1</mn></mstyle></math> to decimal.

The log function can be graphed using the vertical asymptote at <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mo>-</mo><mn>3</mn></mstyle></math> and the points <math><mstyle displaystyle="true"><mrow><mo>(</mo><mo>-</mo><mn>2</mn><mo>,</mo><mo>-</mo><mn>1</mn><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><mo>-</mo><mn>1</mn><mo>,</mo><mo>-</mo><mn>0.13864688</mn><mo>)</mo></mrow></mstyle></math> .

Vertical Asymptote: <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mo>-</mo><mn>3</mn></mstyle></math>

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