Question
What are key feature of a logarithmic parent function
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GPT-4o mini
The logarithmic parent function is \( f(x) = \log_b(x) \), where \( b \) is the base of the logarithm and \( b > 0, b \neq 1 \). Here are some key features of the logarithmic parent function:
1. **Domain**: The domain of the logarithmic function is \( (0, \infty) \). The function is defined only for positive real numbers.
2. **Range**: The range of the logarithmic function is \( (-\infty, \infty) \). It can take any real number as its output.
3. **Intercept**: There is one intercept with the x-axis at \( (1, 0) \) because \( \log_b(1) = 0 \) for any base \( b \).
4. **Asymptote**: The logarithmic function has a vertical asymptote at \( x = 0 \). The function approaches negative infinity as \( x \) approaches zero from the right.
5. **Behavior**: The function is increasing when \( b > 1 \) and decreasing when \( 0 < b < 1 \). For a base greater than 1, as \( x \) increases, \( f(x) \) increases as well.
6. **Shape**: The graph of the logarithmic function rises slowly and never touches the x-axis or the vertical asymptote.
7. **Increasing Nature**: For \( b > 1 \), the logarithmic function is a monotonically increasing function, meaning that as \( x \) increases, \( f(x) \) also increases.
8. **End Behavior**: As \( x \) approaches infinity, \( f(x) \) approaches infinity, while as \( x \) approaches 0 from the right, \( f(x) \) approaches negative infinity.
9. **Transformation**: The basic parent function can undergo transformations, such as shifts and reflections, altering its graph. For example, \( f(x) = \log_b(x - h) + k \) will shift the graph \( h \) units to the right and \( k \) units up.
Understanding these key features can help in analyzing and graphing logarithmic functions in more complex scenarios.
1. **Domain**: The domain of the logarithmic function is \( (0, \infty) \). The function is defined only for positive real numbers.
2. **Range**: The range of the logarithmic function is \( (-\infty, \infty) \). It can take any real number as its output.
3. **Intercept**: There is one intercept with the x-axis at \( (1, 0) \) because \( \log_b(1) = 0 \) for any base \( b \).
4. **Asymptote**: The logarithmic function has a vertical asymptote at \( x = 0 \). The function approaches negative infinity as \( x \) approaches zero from the right.
5. **Behavior**: The function is increasing when \( b > 1 \) and decreasing when \( 0 < b < 1 \). For a base greater than 1, as \( x \) increases, \( f(x) \) increases as well.
6. **Shape**: The graph of the logarithmic function rises slowly and never touches the x-axis or the vertical asymptote.
7. **Increasing Nature**: For \( b > 1 \), the logarithmic function is a monotonically increasing function, meaning that as \( x \) increases, \( f(x) \) also increases.
8. **End Behavior**: As \( x \) approaches infinity, \( f(x) \) approaches infinity, while as \( x \) approaches 0 from the right, \( f(x) \) approaches negative infinity.
9. **Transformation**: The basic parent function can undergo transformations, such as shifts and reflections, altering its graph. For example, \( f(x) = \log_b(x - h) + k \) will shift the graph \( h \) units to the right and \( k \) units up.
Understanding these key features can help in analyzing and graphing logarithmic functions in more complex scenarios.