Does the logarithmic function , where , have turning point(s)? Explain using .
Question
Does the logarithmic function , where , have turning point(s)? Explain using .
Solution
The logarithmic function, f(x) = log_b(x), does not have any turning points.
A turning point is a point at which the derivative of a function changes sign, indicating a local maximum or minimum. For the logarithmic function, the derivative is 1/(x ln(b)), which is always positive for x > 0 and b > 1, and always negative for x > 0 and 0 < b < 1.
Therefore, the function is always increasing for b > 1 and always decreasing for 0 < b < 1. It does not have any points where the derivative changes sign, so it does not have any turning points.
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In mathematics, the logarithm is the inverse function to exponentiation. That means that the logarithm of a number x to the base b is the exponent to which b must be raised to produce x. For example, since 1000 = 103, the logarithm base 10 of 1000 is 3, or log10 (1000) = 3. The logarithm of x to base b is denoted as logb (x), or without parentheses, logb x, or even without the explicit base, log x, when no confusion is possible, or when the base does not matter such as in big O notation.The logarithm base 10 is called the decimal or common logarithm and is commonly used in science and engineering. The natural logarithm has the number e ≈ 2.718 as its base; its use is widespread in mathematics and physics, because of its very simple derivative. The binary logarithm uses base 2 and is frequently used in computer science.
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