Question

In 3 sentences, describe the process for establishing the inverse of a logarithmic function and what will be the resulting function.

Answer 1

Answer:

See below.

Step-by-step explanation:

To establish the inverse of a logarithmic function, first swap the x and y variables in the original function. Then, solve the resulting equation for y, and finally express y as f⁻¹(x). The resulting function will be an exponential function with the same base as the base of the original logarithmic function.

$\dotfill$

As an example, let's find the inverse of y = 2log₃(x - 4).

First, swap the x and y variables:

$\large\text{ x=2\log_3(y-4) }$

Now, solve the equation for y:

$\large\text{ \dfrac{x}{2}=\log_3(y-4) }\\\\\\\large\text{ 3^{\frac{x}{2}}=3^{\log_3(y-4)} }\\\\\\\large\text{ 3^{\frac{x}{2}}=y-4 }\\\\\\\large\text{ y=3^{\frac{x}{2}}+4 }$

Finally, express y as f⁻¹(x):

$\large\text{ f^{-1}(x)=3^{\frac{x}{2}}+4 }$