Answer:
x = -(2 i π n)/(5 log(3)) + 1/5 for n element Z or 1/5
Explanation:
Solve for x:
9^(-x) = 3^(3 x - 1)
Hint: | Take logarithms of both sides to turn products into sums and powers into products.
Take the natural logarithm of both sides and use the identity log(a^b) = b log(a):
-2 log(3) x = log(3) (3 x - 1) + 2 i π n for n element Z
Hint: | Write the linear polynomial on the right-hand side in standard form.
Expand out terms of the right hand side:
-2 log(3) x = -log(3) + 3 log(3) x + 2 i π n for n element Z
Hint: | Isolate x log(3) to the left-hand side.
Subtract 3 x log(3) from both sides:
-5 log(3) x = 2 i π n - log(3) for n element Z
Hint: | Solve for x.
Divide both sides by -5 log(3):
Answer: x = -(2 i π n)/(5 log(3)) + 1/5 for n element Z
_____________________________
Solve for x over the real numbers:
9^(-x) = 3^(3 x - 1)
Hint: | Take logarithms of both sides to turn products into sums and powers into products.
Take the natural logarithm of both sides and use the identity log(a^b) = b log(a):
-2 log(3) x = log(3) (3 x - 1)
Hint: | Divide both sides by a constant to simplify the equation.
Divide both sides by log(3):
-2 x = 3 x - 1
Hint: | Isolate x to the left-hand side.
Subtract 3 x from both sides:
-5 x = -1
Hint: | Solve for x.
Divide both sides by -5:
Answer: x = 1/5