To solve this exponential equation, first let's express the equation T^(x-2) = 49 more clearly. This reads as "T to the power of (x - 2) equals 49".
Next, to simplify this equation, we can apply a logarithm on both sides. Why logarithm? Because logarithms are the inverse operation to exponentiation, and this will therefore help us to remove the exponent in our equation.
Taking natural logarithms (ln) on both sides, we get:
ln(T^(x-2)) = ln(49)
Remember the power rule of logarithms: log(b^n) = n*log(b). This rule lets us to bring down the (x - 2) in front of the logarithm like this:
(x - 2)*ln(T) = ln(49)
Now, our goal is to solve for 'x'. To isolate 'x', we can express the equation as:
x = ln(49)/ln(T) + 2
Let's simplify this further using the value of T. Unfortunately, the value of T is not specified in this problem. To simplify this, let's assume that T = e, where 'e' is the base of the natural logarithm approx 2.71828.
Now, replacing T with e in our equation, we get:
x = ln(49)/ln(e) + 2
We know that ln(e) is essentially 1. Thus, our equation is essentially:
x = ln(49) + 2
Magnificently, using the calculated value of ln(49) and adding 2, we find:
x = 5.891820298110627
Therefore, the solution to our original exponential equation T^(x-2) = 49, where T=e, is approximately x = 5.891820298110627. Please, note that exact x value could be different if T is not equal to e. Always be mindful of your original equation's parameters!