Answer:
x = 1
Step-by-step explanation:
1. Expand the expression: [tex]2^x * 2^3 = 2*2^{-x} + 15[/tex]
2. Evaluate the power: [tex]2^x * 8 = 2 * \frac{1}{2^{x}} +15[/tex]
3. Substitute. Let's use t for [tex]2^x[/tex] : [tex]t * 8 = 2 * \frac{1}{t} +15[/tex]
4. Solve for t :
[tex]8t = \frac{2}{t} + 15[/tex]
[tex]8t - \frac{2}{t} -15 = 0[/tex]
[tex]\frac{8t^2-2-15t}{t}[/tex][tex]= 0[/tex]
[tex]8t^2 -15t -2 = 0[/tex]
[tex]t(8t + 1) -2 (8t + 1) = 0[/tex]
[tex](8t + 1)(t-2) = 0[/tex]
[tex]8t+1 = 0 ; t-2=0[/tex]
[tex]t = -\frac{1}{8}; t=2[/tex]
5. Substitute back:
[tex]2^x =\frac{1}{8} ; 2^x = 2[/tex]
6. Solve for x
[tex]2^x = -\frac{1}{8}[/tex] is false for any value of x because the exponential function is always positive, so there is no real solution.
[tex]2^x = 2^1[/tex] since the bases are the same, set the exponents equal
x = 1
