Understanding the Logarithmic Function | Exploring the Properties and Laws of the log₂x Function

f(x) = log₂x

The function f(x) = log₂x represents a logarithmic function with base 2

The function f(x) = log₂x represents a logarithmic function with base 2. In this case, the input value x must be positive in order for the logarithm to be defined.

The log₂x function calculates the exponent to which 2 must be raised to obtain x. In other words, it answers the question: “What power of 2 gives me x?” The output of the function is the value of that exponent.

For example, let’s evaluate f(x) for x = 4. We want to find the exponent to which 2 must be raised to obtain 4. In this case, 2 raised to the power of 2 equals 4, so f(4) = 2.

Similarly, if we evaluate f(x) for x = 8, we find that 2 raised to the power of 3 equals 8, so f(8) = 3.

The log₂x function has a few key properties:

1. Domain: The domain of the function is the set of all positive real numbers (x > 0). If x is negative or equal to zero, the logarithm is undefined.

2. Range: The range of the function is the set of all real numbers. The function can output any real number as the exponent of 2.

3. Logarithmic Laws: Just like other logarithmic functions, log₂x follows certain laws:

– Product Law: log₂(ab) = log₂a + log₂b. This law states that the logarithm of a product of two numbers is equal to the sum of the logarithms of each individual number.

– Quotient Law: log₂(a/b) = log₂a – log₂b. This law states that the logarithm of the quotient of two numbers is equal to the difference of the logarithms of each individual number.

– Power Law: log₂(a^n) = n * log₂a. This law states that the logarithm of a number raised to a power is equal to the product of the power and the logarithm of the base.

4. Graph: The graph of the function f(x) = log₂x is a curve that starts from the point (1, 0) and increases indefinitely as x approaches positive infinity. The curve never touches or crosses the x-axis.

Remember that when working with logarithmic functions, it is essential to ensure that the input values satisfy the domain restrictions and that the logarithmic laws are applied correctly in calculations.

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