Question

How is the Ancient Puzzle related to the inverse function for y=3^x in problem 5‑53? Show how you can use the idea in the Ancient Puzzle to write an equation in y= form for the inverse y=3^x function of .

Answer 1

To find the inverse function of y=3^x, you take the logarithm with base 3 of both sides, yielding x=log3(y), and then interchange x and y to get the inverse function y=log3(x). This process demonstrates how operations are reversed to solve for the original variable, akin to the ideas presented in the Ancient Puzzle.

Step-by-step explanation:

The Ancient Puzzle, when related to the inverse function for the exponential equation y=3x, emphasizes the process of reversing operations to solve for the original value. Given an exponential function like y=3x, the inverse would be an operation that allows us to find x given a value of y. In the context of exponential functions and their inverses, taking the logarithm of both sides is the method used to achieve this.

In this case, to find the inverse of y=3x, we would take the logarithm with base 3 (logarithm base 3 is denoted as log3), which would give us:

x = log3(y)

Since we need an equation in y= form for the inverse, we simply interchange x and y to get:

y = log3(x)

This equation represents the inverse function of y = 3x, which is crucial since exponential functions and their inverses, such as the natural logarithm (ln) or logarithm base 10, "undo" each other.

Answer 2

The ancient puzzle was created by Indian mathematicians over 2000 years ago. The puzzle can be solved using a simple technique: If loga^b = c, then b^c = a

Step-by-step explanation:

for y = 3^x

To find the inverse, replace y by x and vice-versa, then solve for y

x = 3^y

Take log to base 3 of both sides

log3^x = log3^(3^y)

y = Log 3^x

The inverse of the function y = 3^x given as y = log 3^x completely depicts the idea of the ancient puzzle.