5^(2t+1)-4^(1-t)=0
To solve for t, we can start by adding 4^(1-t) to both sides:
5^(2t+1) = 4^(1-t)
Now, we can take the natural logarithm of both sides to simplify the equation:
ln(5^(2t+1)) = ln(4^(1-t))
Using the property of logarithms that allows us to bring down exponents:
(2t+1)ln(5) = (1-t)ln(4)
Now, we can distribute the ln(5) and ln(4) to simplify further:
2tln(5) + ln(5) = ln(4) - tln(4)
At this point, we can solve for t by isolating the t term:
2tln(5) + tln(4) = ln(4) - ln(5)
Factor out the t:
t(2ln(5) - ln(4)) = ln(4) - ln(5)
Finally, divide both sides by (2ln(5) - ln(4)) to solve for t:
t = (ln(4) - ln(5)) / (2ln(5) - ln(4))
This is the solution for t.