## natural log function

### The natural logarithm function, denoted as ln(x) or loge(x), is the inverse of the exponential function with base e (Euler’s number)

The natural logarithm function, denoted as ln(x) or loge(x), is the inverse of the exponential function with base e (Euler’s number). In other words, if y = e^x, then x = ln(y).

The natural logarithm function is defined for positive real numbers. It is a logarithmic function that describes the time it takes for a quantity to decay or grow exponentially. It is widely used in various branches of mathematics, science, and engineering.

Properties of the natural logarithm function include:

1. Domain: The natural logarithm function is defined for positive real numbers, so the domain is (0, ∞).

2. Range: The range of the natural logarithm function is (-∞, ∞), since logarithms can be positive or negative.

3. Basic Property: The natural logarithm of 1 is 0, so ln(1) = 0.

4. Logarithmic Product Rule: ln(xy) = ln(x) + ln(y) for any positive real numbers x and y.

5. Logarithmic Quotient Rule: ln(x/y) = ln(x) – ln(y) for any positive real numbers x and y.

6. Logarithmic Power Rule: ln(x^a) = a ln(x) for any positive real number x and real number a.

7. Logarithmic Change of Base Formula: ln(x) = logb(x) / logb(e), where logb(x) represents the logarithm of x with base b.

The natural logarithm function is useful in calculus, particularly in integration and solving differential equations. It is also used in exponential growth and decay problems, probability and statistics, and various scientific and economic models.

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