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Trigonometry Examples
Step 1
Step 1.1
Set the argument of the logarithm equal to zero.
Step 1.2
Solve for .
Step 1.2.1
Factor the left side of the equation.
Step 1.2.1.1
Factor out of .
Step 1.2.1.1.1
Factor out of .
Step 1.2.1.1.2
Factor out of .
Step 1.2.1.1.3
Factor out of .
Step 1.2.1.1.4
Factor out of .
Step 1.2.1.1.5
Factor out of .
Step 1.2.1.2
Factor using the perfect square rule.
Step 1.2.1.2.1
Rewrite as .
Step 1.2.1.2.2
Check that the middle term is two times the product of the numbers being squared in the first term and third term.
Step 1.2.1.2.3
Rewrite the polynomial.
Step 1.2.1.2.4
Factor using the perfect square trinomial rule , where and .
Step 1.2.2
Divide each term in by and simplify.
Step 1.2.2.1
Divide each term in by .
Step 1.2.2.2
Simplify the left side.
Step 1.2.2.2.1
Cancel the common factor of .
Step 1.2.2.2.1.1
Cancel the common factor.
Step 1.2.2.2.1.2
Divide by .
Step 1.2.2.3
Simplify the right side.
Step 1.2.2.3.1
Divide by .
Step 1.2.3
Set the equal to .
Step 1.2.4
Subtract from both sides of the equation.
Step 1.3
The vertical asymptote occurs at .
Vertical Asymptote:
Vertical Asymptote:
Step 2
Step 2.1
Replace the variable with in the expression.
Step 2.2
Simplify the result.
Step 2.2.1
Simplify each term.
Step 2.2.1.1
Raise to the power of .
Step 2.2.1.2
Multiply by .
Step 2.2.1.3
Multiply by .
Step 2.2.2
Simplify by adding and subtracting.
Step 2.2.2.1
Subtract from .
Step 2.2.2.2
Add and .
Step 2.2.3
Logarithm base of is .
Step 2.2.4
The final answer is .
Step 2.3
Convert to decimal.
Step 3
Step 3.1
Replace the variable with in the expression.
Step 3.2
Simplify the result.
Step 3.2.1
Simplify each term.
Step 3.2.1.1
Raise to the power of .
Step 3.2.1.2
Multiply by .
Step 3.2.1.3
Multiply by .
Step 3.2.2
Simplify by adding and subtracting.
Step 3.2.2.1
Subtract from .
Step 3.2.2.2
Add and .
Step 3.2.3
Logarithm base of is .
Step 3.2.3.1
Rewrite as an equation.
Step 3.2.3.2
Rewrite in exponential form using the definition of a logarithm. If and are positive real numbers and does not equal , then is equivalent to .
Step 3.2.3.3
Create equivalent expressions in the equation that all have equal bases.
Step 3.2.3.4
Since the bases are the same, the two expressions are only equal if the exponents are also equal.
Step 3.2.3.5
The variable is equal to .
Step 3.2.4
The final answer is .
Step 3.3
Convert to decimal.
Step 4
Step 4.1
Replace the variable with in the expression.
Step 4.2
Simplify the result.
Step 4.2.1
Simplify each term.
Step 4.2.1.1
Raise to the power of .
Step 4.2.1.2
Multiply by .
Step 4.2.1.3
Multiply by .
Step 4.2.2
Simplify by adding and subtracting.
Step 4.2.2.1
Subtract from .
Step 4.2.2.2
Add and .
Step 4.2.3
Logarithm base of is .
Step 4.2.3.1
Rewrite as an equation.
Step 4.2.3.2
Rewrite in exponential form using the definition of a logarithm. If and are positive real numbers and does not equal , then is equivalent to .
Step 4.2.3.3
Create equivalent expressions in the equation that all have equal bases.
Step 4.2.3.4
Since the bases are the same, the two expressions are only equal if the exponents are also equal.
Step 4.2.3.5
The variable is equal to .
Step 4.2.4
The final answer is .
Step 4.3
Convert to decimal.
Step 5
The log function can be graphed using the vertical asymptote at and the points .
Vertical Asymptote:
Step 6