If 3 to the exponent negative x is equals to 4t, express 4 to the exponent x minus 1 plus 4 to the exponent x plus 1 all over 17 times 12 to the exponent x in terms of t?

3^-x = 4t

3^x = 1/(4t)

Do you mean

4^(x-1) + 4^(x+1)
--------------------
17 * 12^x

or some other grouping? All those words just cloud the issue.

There's really no good way to mix the 3's and 4's as bases. I suspect a typo.

Did you maybe mean

3^-x = 4^t

??

To solve this problem, we need to manipulate the given equation and express the desired expression in terms of t. Let's break it down step by step:

Given equation: 3^(-x) = 4t

Step 1: Start by isolating x. Take the logarithm of both sides of the equation. We'll use the natural logarithm (ln) for convenience:

ln(3^(-x)) = ln(4t)

Step 2: Use the logarithmic property that ln(a^b) = b * ln(a):

-x * ln(3) = ln(4t)

Step 3: Divide both sides by -ln(3) to solve for x:

x = ln(4t) / -ln(3)

Step 4: Now, let's express the desired expression in terms of t:

4^x-1 + 4^x+1 / (17 * 12^x)

Step 5: Substitute the value of x obtained in Step 3:

4^(ln(4t) / -ln(3)) - 1 + 4^(ln(4t) / -ln(3)) + 1 / (17 * 12^(ln(4t) / -ln(3)))

Step 6: Simplify the expression further:

Using the property that a^(b+c) = a^b * a^c, we can separate the exponents:

4^ln(4t) / -ln(3) * (4^(-1) + 4^1) / (17 * 12^ln(4t) / -ln(3))

Step 7: Simplify the exponents:

4^ln(4t) / -ln(3) * (1/4 + 4) / (17 * 12^ln(4t) / -ln(3))

Step 8: Simplify the fractions:

4^ln(4t) / -ln(3) * (17/4) / (17 * 12^ln(4t) / -ln(3))

Step 9: Simplify further:

The exponent -ln(3) in the denominator cancels out with the exponent ln(4t) in the numerator:

4^ln(4t) * (17/4) / (17 * 12^ln(4t))

Step 10: Finally, rearrange the expression:

(17/4) * 4^ln(4t) / (17 * 12^ln(4t))

Therefore, the expression 4^x-1 + 4^x+1 / (17 * 12^x) in terms of t is (17/4) * 4^ln(4t) / (17 * 12^ln(4t)).

Please note that this solution assumes that the exponents are only meant for the terms immediately following them. If this assumption is incorrect, please provide more explicit parentheses in your equation.