I need to solve for t in the following expression.

A/(1+B^t/2) = C using logarithms. how do I get started

To solve for t in the equation A/(1+B^t/2) = C using logarithms, you can follow these steps:

Step 1: Rewrite the equation using base-10 logarithms.
Take the logarithm of both sides of the equation, using any base you prefer, commonly base-10 (log) or base-e (ln). For simplicity, we will use base-10 logarithms (log).

log(A/(1+B^t/2)) = log(C)

Step 2: Apply properties of logarithms.
Using the properties of logarithms, we can rewrite the equation further.
- The logarithm of a quotient is equal to the difference of logarithms.
- The logarithm of a power is equal to the product of the logarithm.

log(A) - log(1+B^t/2) = log(C)

Step 3: Simplify the equation.
We can simplify further by using the logarithmic rule of logarithm of 1, which is equal to 0.

log(A) - log(1+B^t/2) = 0

Step 4: Apply the inverse logarithm function.
Apply the inverse function of logarithm, which is exponentiation, to eliminate the logarithm on the left side of the equation.

10^(log(A) - log(1+B^t/2)) = 10^0
10^(log(A)) / 10^(log(1+B^t/2)) = 1
A / (1 + B^(t/2)) = 1

Step 5: Solve for t.
Now, we have a simpler equation after eliminating the logarithms. Rearrange the equation to isolate (1 + B^(t/2)).

A / (1 + B^(t/2)) = 1
A = 1 + B^(t/2)
1 = A - B^(t/2)
B^(t/2) = A - 1

Step 6: Continue solving for t.
To solve for t, we need to isolate the base B. Take the logarithm of both sides of the equation using the same base B, typically log base B.

log_B(B^(t/2)) = log_B(A - 1)
t/2 = log_B(A - 1)

Step 7: Multiply by 2.
Multiply both sides of the equation by 2 to solve for t.

t = 2 * log_B(A - 1)

Therefore, the solution for t in the equation A/(1+B^t/2) = C, using logarithms, is t = 2 * log_B(A - 1).