Hint: use logarithms:
Hello there! I was just wondering if someone could kindly show me how to do the following questions:
1) 4^x + 6(4^-x) = 5
2) The half-life of a certain substance is 3.6 days. How long will it take for 20g of the substance to decay to 7g?
-The only equation I know of is y = ab^x but I'm not sure what to put where, thank you!!
Hello, Marmalade!
You can derive the half-life formula yourself . . .2) The half-life of a certain substance is 3.6 days.
How long will it take for 20g of the substance to decay to 7g?
The only equation I know of is: .
but I'm not sure what to put where, thank you!
The general form is: . .[1]
. . where: .
We are given: .
Substitute into [1]: .
Take logs: .
. . Then: .
Hence, the function is: .
For this problem,
Now we can answer the question: When will ?
We have: .
. . days
Another way to do this is to recognize that "half" life means that the amount is multiplied by 1/2 every 3.6 days- In T days, there are T/3.6 periods of 3.6 days each which means the initial amount is multiplied by 1/2 T/3.6 times: . You want to solve or, dividing both sides by 20, . To solve that take the logarithm of both sides (it doesn't matter whether you take the natural logarithm or common logarithm) so [tex]T= 3.6\frac{log(7/20)}{log(1/2)}= 3.6\frac{log(20)- log(7)}{log(2)}, exactly what Soroban got.
The moral is that all logarithms and all exponentials are equivalent. It doesn't matter what base you use so use whatever is simplest.