solve the exponential equation step by step
1/16=64^4x-3
To solve the exponential equation 1/16 = 64^(4x-3) step by step, we can follow these steps:
Step 1: Rewrite the equation using exponential notation.
1/16 = (64^(4x-3))^-1
Step 2: Apply the rule of exponents:
1/16 = 64^(-4x+3)
Step 3: Convert the equation to a common base by writing 64 as a power of 4:
1/16 = (4^3)^(-4x+3)
Step 4: Simplify the right side of the equation:
1/16 = 4^(3*(-4x+3))
Step 5: Apply the rule of exponents to the right side of the equation:
1/16 = 4^(-12x+9)
Step 6: Rewrite 1/16 as a power of 4:
4^(-2) = 4^(-12x+9)
Step 7: Set the exponents equal to each other since the bases are the same:
-2 = -12x + 9
Step 8: Move the constants to one side of the equation and the variable to the other side:
12x = 9 + 2
Step 9: Simplify:
12x = 11
Step 10: Divide both sides by 12 to solve for x:
x = 11/12
Therefore, the solution to the exponential equation 1/16 = 64^(4x-3) is x = 11/12.