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Questions and Answers
Which of the following is the correct domain for the logarithmic function $f(x) = ext{log}(x-3)$?
Which of the following is the correct domain for the logarithmic function $f(x) = ext{log}(x-3)$?
- $x < 3$
- $x > 3$ (correct)
- All real numbers
- $x \geq 3$
What is the range of the function $g(x) = ext{log}(x)$?
What is the range of the function $g(x) = ext{log}(x)$?
- Negative real numbers
- Non-negative real numbers
- Positive real numbers
- All real numbers (correct)
When condensing the expression $3 ext{log}(x) + 2 ext{log}(y)$, which of the following is the correct form?
When condensing the expression $3 ext{log}(x) + 2 ext{log}(y)$, which of the following is the correct form?
- log($3xy^2$)
- log($x^3 y^2$) (correct)
- log($x^2y^3$)
- log($xy^3)$
What is the horizontal asymptote of the logarithmic function $h(x) = ext{log}(x)$?
What is the horizontal asymptote of the logarithmic function $h(x) = ext{log}(x)$?
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Which equation correctly represents solving the logarithmic equation $\text{log}_2(x) = 3$?
Which equation correctly represents solving the logarithmic equation $\text{log}_2(x) = 3$?
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Study Notes
Asymptotes, Domain, and Range of Logarithmic Functions
- Logarithmic functions have vertical asymptotes where the argument of the logarithm equals zero.
- The general form of a logarithmic function is f(x) = log_b(x - h) + k, where (h, k) translates the graph.
- The domain of a logarithmic function is x > h; it includes all x-values greater than the horizontal shift.
- The range of a logarithmic function is all real numbers, (-∞, ∞), indicating it can take any vertical value.
Expanding and Condensing Logarithmic Functions
- To expand logarithmic expressions, use properties such as:
- log_b(MN) = log_b(M) + log_b(N)
- log_b(M/N) = log_b(M) - log_b(N)
- log_b(M^p) = p * log_b(M)
- Condensing logarithmic expressions involves combining logs using the above properties.
- Example of condensing: log_b(M) + log_b(N) = log_b(MN)
Solving Logarithmic Equations
- To solve logarithmic equations, if possible, rewrite the equation in exponential form: if log_b(x) = y, then b^y = x.
- Check for extraneous solutions, especially when manipulating logarithmic expressions.
- Common strategies include isolating the logarithm, using properties of logarithms, and applying one-to-one properties of logarithmic functions.
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