The Logarithmic Function F(X) = Log₂X: Properties, Behavior, And Graph

f(x) = log₂x

Base-2 log

let’s take a look at the function f(x) = log₂x.

To start, let’s define some important concepts related to logarithmic functions.

A logarithm is the power to which a base must be raised to obtain a particular number. In the case of f(x) = log₂x, the base is 2 and the result is x. That is, log₂x tells us the power to which we must raise 2 to obtain x.

One important property of logarithms is that they can be used to rewrite exponential expressions. Specifically, if we have an exponential expression of the form a^b = c, we can rewrite it as logₐc = b.

Now, let’s analyze some key characteristics of the function f(x) = log₂x.

Domain: The domain of a logarithmic function is all positive real numbers, since we cannot take the logarithm of a negative number. So, the domain of f(x) = log₂x is (0, ∞).

Range: The range of f(x) = log₂x is all real numbers. This means that for any real number y, there is a positive value of x that satisfies the equation y = log₂x.

x-Intercept: The x-intercept occurs when y = 0. So, we can set log₂x = 0 and solve for x:

log₂x = 0
2⁰ = x
x = 1

Therefore, the x-intercept of f(x) = log₂x is (1, 0).

y-Intercept: To find the y-intercept, we set x = 1:

y = log₂(1)
y = 0

So, the y-intercept of f(x) = log₂x is (0, 0).

Symmetry: Logarithmic functions are not symmetric about the x-axis, y-axis, or origin.

Behavior: As x approaches 0 from the right side (i.e. x → 0⁺), log₂x approaches negative infinity. As x approaches positive infinity (i.e. x → ∞), log₂x approaches positive infinity.

Graph: The graph of f(x) = log₂x looks like a curved line that approaches the x-axis as x approaches infinity and increases quickly (but smoothly) as x increases.

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