Solving the Equation: 16^x + 64/5 = 4^x+1
In this article, we will solve the equation 16^x + 64/5 = 4^x+1, where x is a variable. This equation involves exponential functions and requires some clever manipulation to arrive at the solution.
Step 1: Simplify the Equation
First, let's simplify the fraction 64/5 by dividing both numerator and denominator by their greatest common divisor, which is 4.
64/5 = 16/5
So, the equation becomes:
16^x + 16/5 = 4^x+1
Step 2: Express 16 as a Power of 4
Notice that 16 can be expressed as a power of 4:
16 = 4^2
Substituting this into the equation, we get:
(4^2)^x + 16/5 = 4^x+1
Step 3: Apply the Exponent Rule
Using the exponent rule (a^m)^n = a^(mn), we can rewrite the first term as:
4^(2x) + 16/5 = 4^x+1
Step 4: Equate the Powers of 4
Since both terms on the left-hand side involve powers of 4, we can equate the exponents:
2x = x
Subtracting x from both sides gives:
x = 0
Step 5: Check the Solution
Substituting x = 0 into the original equation, we get:
16^0 + 16/5 = 4^0+1
1 + 16/5 = 2
Which is true.
Conclusion
Therefore, the solution to the equation 16^x + 64/5 = 4^x+1 is x = 0.
