Trigonometry Examples

Graph f(x) = log base 3 of 3x
Step 1
Find the asymptotes.
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Step 1.1
Set the argument of the logarithm equal to zero.
Step 1.2
Divide each term in by and simplify.
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Step 1.2.1
Divide each term in by .
Step 1.2.2
Simplify the left side.
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Step 1.2.2.1
Cancel the common factor of .
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Step 1.2.2.1.1
Cancel the common factor.
Step 1.2.2.1.2
Divide by .
Step 1.2.3
Simplify the right side.
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Step 1.2.3.1
Divide by .
Step 1.3
The vertical asymptote occurs at .
Vertical Asymptote:
Vertical Asymptote:
Step 2
Find the point at .
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Step 2.1
Replace the variable with in the expression.
Step 2.2
Simplify the result.
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Step 2.2.1
Multiply by .
Step 2.2.2
Logarithm base of is .
Step 2.2.3
The final answer is .
Step 2.3
Convert to decimal.
Step 3
Find the point at .
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Step 3.1
Replace the variable with in the expression.
Step 3.2
Simplify the result.
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Step 3.2.1
Multiply by .
Step 3.2.2
Logarithm base of is .
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Step 3.2.2.1
Rewrite as an equation.
Step 3.2.2.2
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 3.2.2.3
Create equivalent expressions in the equation that all have equal bases.
Step 3.2.2.4
Since the bases are the same, the two expressions are only equal if the exponents are also equal.
Step 3.2.2.5
The variable is equal to .
Step 3.2.3
The final answer is .
Step 3.3
Convert to decimal.
Step 4
Find the point at .
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Step 4.1
Replace the variable with in the expression.
Step 4.2
Simplify the result.
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Step 4.2.1
Multiply by .
Step 4.2.2
Logarithm base of is .
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Step 4.2.2.1
Rewrite as an equation.
Step 4.2.2.2
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 4.2.2.3
Create equivalent expressions in the equation that all have equal bases.
Step 4.2.2.4
Since the bases are the same, the two expressions are only equal if the exponents are also equal.
Step 4.2.2.5
The variable is equal to .
Step 4.2.3
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