Understanding the Inverse Relationship between Exponential and Logarithmic Functions

the inverse of the exponential function, given as f(x)=logax

The inverse of the exponential function f(x) = a^x, where a is a positive constant greater than 1, is the logarithmic function g(x) = log_a(x)

The inverse of the exponential function f(x) = a^x, where a is a positive constant greater than 1, is the logarithmic function g(x) = log_a(x).

To understand how these functions are inverses of each other, let’s consider an example. Let’s assume a = 2. For the exponential function f(x) = 2^x, if we plug in x = 2, we get f(2) = 2^2 = 4. This tells us that when we raise 2 to the power of 2, we get 4.

Now, if we want to find the inverse of this function, we switch the x and y variables. So, instead of y = 2^x, we have x = 2^y. Solving for y in terms of x, we take the logarithm of both sides with base 2. This gives us log_2(x) = y. Here, x is the input and y is the output of the inverse function.

Therefore, the inverse of f(x) = 2^x is g(x) = log_2(x). If we plug in x = 4 into the inverse function, we get g(4) = log_2(4) = 2. This tells us that if we raise 2 to the power of 2, we get 4, just like we observed when evaluating f(2) = 4.

In summary, the inverse of an exponential function of the form f(x) = a^x is the logarithmic function of the same base, given by g(x) = log_a(x). This applies to any positive constant a greater than 1.

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