Question

Find the inverse without using natural log
A(t)=500(1/2)^(t/272)

Answer 1

The inverse of A(t) = 500(1/2)^(t/272) can be found by isolating t, resulting in t(y) = -272 × log2(y/500), assuming the use of log base 2 is allowed instead of natural logarithms.

Step-by-step explanation:

To find the inverse of the function A(t) = 500(1/2)^(t/272) without using the natural logarithm, follow these steps:

1. 
   
3. Consider the original equation and replace A(t) with y for convenience: y = 500(1/2)^(t/272).
4. 
   
6. To find the inverse, solve for t in terms of y. Start by dividing both sides by 500: y/500 = (1/2)^(t/272).
7. 
   
9. Recognize that (1/2) is 2^(-1) and rewrite the equation: y/500 = 2^(-t/272).
10. 
    
12. To undo the power of 2, we want to use logarithms. However, since we're not using natural logs, consider the property that log base b of b to the power of x is x: let's apply log base 2 to both sides, leading to log2(y/500) = -t/272.
13. 
    
15. Multiply both sides by -272 to isolate t: t = -272 × log2(y/500).
16. 
    
18. Since the question forbids using logarithms, you would typically be stuck here. However, if you're allowed to use log base 2, the final form of the inverse function is t(y) = -272 × log2(y/500).

Note that finding the inverse function usually requires logarithms, but if using base 2 logarithms is permissible, then it is possible to solve for t without using the natural logarithm (ln).