To solve the equation [tex]\( 81^{1 - 3x} = 9^{-4x} \)[/tex], let's proceed step by step.
### Step 1: Express Both Sides Using the Same Base
First, it helps to express the numbers on both sides of the equation in terms of the same base. Let's note that:
- [tex]\( 81 = 3^4 \)[/tex]
- [tex]\( 9 = 3^2 \)[/tex]
### Step 2: Rewrite the Equation
Rewrite [tex]\( 81 \)[/tex] and [tex]\( 9 \)[/tex] in terms of base [tex]\( 3 \)[/tex]:
[tex]\[ 81^{1 - 3x} = (3^4)^{1 - 3x} \][/tex]
[tex]\[ 9^{-4x} = (3^2)^{-4x} \][/tex]
### Step 3: Apply the Exponent Rules
Recall that [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]. Applying this, we get:
[tex]\[ (3^4)^{1 - 3x} = 3^{4(1 - 3x)} \][/tex]
[tex]\[ (3^2)^{-4x} = 3^{2(-4x)} \][/tex]
Thus, the equation becomes:
[tex]\[ 3^{4(1 - 3x)} = 3^{2(-4x)} \][/tex]
### Step 4: Set Exponents Equal to Each Other
Since the bases (base [tex]\( 3 \)[/tex]) are the same, we can set the exponents equal:
[tex]\[ 4(1 - 3x) = 2(-4x) \][/tex]
### Step 5: Simplify the Exponents
Distribute the constants in the exponents:
[tex]\[ 4 - 12x = -8x \][/tex]
### Step 6: Solve for [tex]\( x \)[/tex]
Rearrange the equation to isolate [tex]\( x \)[/tex]:
[tex]\[ 4 - 12x = -8x \][/tex]
[tex]\[ 4 = -8x + 12x \][/tex]
[tex]\[ 4 = 4x \][/tex]
[tex]\[ x = 1 \][/tex]
### Additional Complex Solutions
We also need to consider solutions in the complex domain. The equation [tex]\( 81^{1 - 3x} = 9^{-4x} \)[/tex] may have more complex solutions. Taking into account the properties of exponentials and logarithms, along with the periodic nature of complex logarithms, the complete solutions set includes additional complex numbers:
[tex]\[ x = 1 \][/tex]
[tex]\[ x = \frac{\log(9) - i\pi}{2\log(3)} \][/tex]
[tex]\[ x = \frac{\log(9) + i\pi}{2\log(3)} \][/tex]
[tex]\[ x = 1 + \frac{i\pi}{\log(3)} \][/tex]
This gives us a complete set of solutions:
[tex]\[ x = 1 \][/tex]
[tex]\[ x = \frac{\log(9) - i\pi}{2\log(3)} \][/tex]
[tex]\[ x = \frac{\log(9) + i\pi}{2\log(3)} \][/tex]
[tex]\[ x = 1 + \frac{i\pi}{\log(3)} \][/tex]
These are the solutions to the given equation [tex]\( 81^{1 - 3x} = 9^{-4x} \)[/tex].
