To solve for x in the equation 16^x + 3 = 4^ (x+1), follow these steps:
Isolate the x terms on one side of the equation by subtracting 3 from both sides:
16^x + 3 - 3 = 4^ (x+1) - 3
16^x = 4^ (x+1) - 3
Take the logarithm of both sides to get rid of the exponential functions:
log16 (16^x) = log16 (4^ (x+1) - 3)
x = log16 (4^ (x+1) - 3) / log16 (16)
Use the logarithmic identity logb (a^c) = c * logb (a) on the right side:
x = (x + 1) * log16 (4) / log16 (16)
Simplify the right side by using logarithmic identities:
x = (x + 1) * log16 (2^2) / log16 (2^4)
x = (x + 1) * 2 / 4
Solve for x by dividing both sides by x + 1 and multiplying by 4:
x * 4 / (x + 1) = 2
4x = 2x + 2
2x = 2
x = 1
Therefore, the solution for x is x = 1.