Final answer:
To solve the equation 3x^(1+2x) = 4^(x-1), you can start by taking the natural logarithm (ln) of both sides. Then simplify the equation by substituting a variable and raising e (the base of natural logarithm) to both sides. Finally, divide by 3 to find the exact solution for x.
Step-by-step explanation:
To solve the equation 3x^(1+2x) = 4^(x-1), we can start by taking the natural logarithm (ln) of both sides:
ln(3x^(1+2x)) = ln(4^(x-1))
Apply the logarithmic property: ln(a^b) = b*ln(a)
(1+2x) * ln(3x) = (x-1) * ln(4)
Now let's simplify this equation: ln(3x) + 2x*ln(3x) = x*ln(4) - ln(4)
We can substitute y = 3x and rewrite the equation as: ln(y) + 2/3 * ln(y) = 1/3 * ln(4) - ln(4)
3/3 * ln(y) + 2/3 * ln(y) = 1/3 * ln(4) - ln(4)
ln(y) = 1/3 * ln(4) - ln(4)
Combine the terms: ln(y) = ln(4) * (1/3 - 1)
ln(y) = ln(4) * (-2/3)
Now raise e (the base of natural logarithm) to both sides:
y = e^(ln(4) * (-2/3))
Finally, substitute y back into our equation: 3x = e^(ln(4) * (-2/3))
Divide both sides by 3: x = e^(ln(4) * (-2/3)) / 3