Question

Solve the exponential equation for x 3^7x+4=(1/27)^x-3

Answer 1

Answer:

The solution is $x = \frac{1}{2}$

Step-by-step explanation:

To solve this exponential equation, we must write both sides as powers of 3.

The equation given is:

$3^{7x + 4} = (\frac{1}{27})^{x-3}$

Since $3^3 = 27$, we have that $3^{-3} = \frac{1}{3^3} = \frac{1}{27}$, so$\frac{1}{27} = 3^{-3}$

Then

$3^{7x + 4} = (\frac{1}{27})^{x-3}$

$3^{7x + 4} = (3^{-3})^{x-3}$

$3^{7x + 4} = 3^{-3(x-3)}$

$3^{7x + 4} = 3^{-3x + 9}$

Since both sides are now powers of 3, we can equal them:

$7x + 4 = -3x + 9$

$10x = 5$

$x = \frac{5}{10}$

$x = \frac{1}{2}$

The solution is $x = \frac{1}{2}$