How would I figure this out:
4^t = 8^t-1
Just by looking at it, I know that x = 3 but how would I figure this out mathematically?
4^t = (2^2)^t = 2^(2t)
8^(t-1) = (2^3)^(t-1) = 2^(3t-3)
so 2^(3t-3) = 2^(2t)
then 3t-3 = 2t
t = 3
Thanks so much :)
To solve the equation 4^t = 8^(t-1) mathematically, we can start by rewriting 8 as 2^3:
4^t = (2^3)^(t-1)
Next, we can apply the exponent rule of raising a power to a power, which states that (a^m)^n = a^(m*n):
4^t = 2^(3*(t-1))
Now, we can simplify the equation further:
4^t = 2^(3t - 3)
To make the bases on both sides of the equation the same, we can rewrite 4 as 2^2:
(2^2)^t = 2^(3t - 3)
Using the same exponent rule as before, we can simplify:
2^(2t) = 2^(3t - 3)
Since the bases are the same, we can equate the exponents:
2t = 3t - 3
Now, we can solve for t by moving all the terms involving t to one side of the equation:
2t - 3t = -3
Simplifying further:
-t = -3
Finally, multiplying both sides of the equation by -1 yields:
t = 3
Therefore, the value of t that satisfies the equation 4^t = 8^(t-1) is t = 3.