1/9^x=3^-x/3×81^x
find the value of x

x = -(2 i π n)/(5 log(3)) + 1/5 for n element Z or 1/5

Step-by-step 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):

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:

x = 1/5

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step-by-step explanation: