7^t-2 = 5^t
Do you need to solve for t?
Hint - take logs.
yes
is it 7^(t-2) = 5^t or
7^t - 2 = 5^t ?
If it is the first way, then
log(7^(t-2)) = log 5^t
(t-2)log7 = tlog5
tlog7 - 2lo7 = tlog5
tlog7 - tlog5 = 2log7
t(log7 - log5) - 2log7
t = 2log7 / (log7 - log5)
t = 11.566542
for the second interpretation of your typing, one would have to use something like Newton's Method to solve it.
To solve the equation 7^t-2 = 5^t, let's take the logarithm of both sides of the equation to help us get rid of the exponents.
1. Start by taking the logarithm to the base 10 of both sides of the equation:
log(7^t-2) = log(5^t)
2. Use the logarithmic property log(a^b) = b * log(a) to simplify the equation further:
(t - 2) * log(7) = t * log(5)
3. Distribute the term on the left side of the equation:
t * log(7) - 2 * log(7) = t * log(5)
4. Group the terms with t on one side of the equation:
t * log(7) - t * log(5) = 2 * log(7)
5. Factor out t from the left side of the equation:
t * (log(7) - log(5)) = 2 * log(7)
6. Divide both sides of the equation by (log(7) - log(5)):
t = (2 * log(7)) / (log(7) - log(5))
Now we have the solution for t in terms of logarithms. You can use a calculator to evaluate the expression on the right side of the equation and get the numerical value of t.