Solved on Nov 01, 2023

Solve the equation (e3)2x8=e4x+5(e^{3})^{2x-8} = e^{-4x+5}

STEP 1

Assumptions1. The base of the exponential functions is ee, the natural logarithm base. . The equation is (e3)x8=e4x+5\left(e^{3}\right)^{ x-8}=e^{-4 x+5}

STEP 2

First, we simplify the equation. The power of a power rule states that (am)n=amn(a^m)^n = a^{mn}. So, we can simplify (e)2x8\left(e^{}\right)^{2 x-8} to e(2x8)e^{(2x-8)}.
(e)2x8=e(2x8)\left(e^{}\right)^{2 x-8} = e^{(2x-8)}

STEP 3

Now, we simplify the expression 3(2x8)3(2x-8) to 6x246x-24.
e3(2x8)=e6x24e^{3(2x-8)} = e^{6x-24}

STEP 4

Now we have the equation e6x24=e4x+e^{6x-24}=e^{-4 x+}.

STEP 5

Since the bases are the same, we can equate the exponents. This gives us the equation x24=4x+5x-24=-4x+5.
x24=4x+5x-24=-4x+5

STEP 6

Now, we solve the equation for xx. First, we add 4x4x to both sides of the equation to get 10x24=510x-24=5.
10x24=510x-24=5

STEP 7

Next, we add24 to both sides of the equation to get 10x=2910x=29.
10x=2910x=29

STEP 8

Finally, we divide both sides of the equation by10 to solve for xx.x=2910x=\frac{29}{10}The solution to the equation (e3)2x8=e4x+5\left(e^{3}\right)^{2 x-8}=e^{-4 x+5} is x=2910x=\frac{29}{10}.

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