Solved: "3^(2x) - 7(3^x) - 18 = 0"

College

Demoy Levene

Answered question

2022-10-16

3^(2x) - 7(3^x) - 18 = 0

Answer & Explanation

nick1337

Expert2023-06-17Added 777 answers

To solve the equation

3^{2 x} - 7 (3^{x}) - 18 = 0

for

x

, we can make a substitution to simplify the equation.
Let's make a substitution:

u = 3^{x}

.
Now, we can rewrite the equation in terms of

u

u^{2} - 7 u - 18 = 0

To solve this quadratic equation, we can factor it:

(u - 9) (u + 2) = 0

Now, we can solve for

u

u - 9 = 0 or u + 2 = 0

u = 9 or u = - 2

Since

u = 3^{x}

, we can substitute back to find the values of

x

3^{x} = 9 or 3^{x} = - 2

For the equation

3^{x} = 9

, we can take the logarithm base 3 of both sides:

x = \log_{3} (9)

x = 2

For the equation

3^{x} = - 2

, there are no real solutions, as

3^{x}

is always positive.
Therefore, the solution to the equation

3^{2 x} - 7 (3^{x}) - 18 = 0

x = 2

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