Definition:
The logarithmic function is defined as:
\[ f(x) = \log_a(x) \]
where, \( a \) is the base of the logarithm, and it must satisfy \( a > 0 \) and \( a \neq 1 \). The function is the inverse of the exponential function \( y = a^x \), meaning it answers the question: "To what exponent must \( a \) be raised to obtain \( x \)?"
Note: If the base of the logarithmic function a = 10, then it is called the
common logarithmic function, which is the default value.
Properties Based on Base \(( a )\)
-
Case: \( a > 1 \)
- The function \( f(x) = \log_a(x) \) is increasing, meaning that as \( x \) increases, \( f(x) \) also increases.
- Example:
If \( a = 2 \), then \( \log_2(8) = 3 \) because \( 2^3 = 8 \).
- Domain: \( (0, \infty) \) (i.e., \( x \) must be positive)
- Range: \( (-\infty, \infty) \) (i.e., the function can take all real values).
- x-intercept: (1,0); and y-intercept: none.
- The graph of the logarithmic function is smooth and continuous, containing the points: \((\frac{1}{a}\), - 1), \((1, 0)\), and \((a, 1)\).
-
Case: \( 0< a< 1 \)
- The function \( f(x) = \log_a(x) \) is decreasing, meaning that as \( x \) increases, \( f(x) \) decreases.
- Example:
If \( a = \frac{1}{2} \), then \( \log_{\frac{1}{2}}(8) = -3 \) because \( (\frac{1}{2})^{-3} = 8 \).
- Domain: \( (0, \infty) \) (same as before)
- Range: \( (-\infty, \infty) \) (same as before)
- x-intercept: (1,0); and y-intercept: none.
- The graph of the logarithmic function is smooth and continuous, containing the points: \((a, 1)\) , \((1, 0)\), and \((\frac{1}{a}\), - 1).
Graphical Interpretation
- When \( a > 1 \), the graph of \( f(x) = \log_a(x) \) rises from left to right.
- It has a vertical asymptote at \( x = 0 \), meaning it approaches negative infinity as \( x \) approaches zero from the right.
- When \( 0 < a < 1 \), the graph declines from left to right.
- It has a vertical asymptote at \( x = 0 \), meaning it approaches positive infinity as \( x \) approaches zero from the right.
- In both cases, the function crosses the x-axis at ( x = 1 ) because \( \log_a(1) = 0 \).