Answer:
[tex]x = 7[/tex]
Step-by-step explanation:
1) Use Division Distributive Property: (x/y)^a = x^a/y^a.
[tex] \frac{1}{ {27}^{x - 8} } = 27[/tex]
2) Multiply both sides by 27^x - 8.
[tex]1 = {27} \times 27^{1 + x - 8} [/tex]
3) Use the product rule: x^a x^b = x^a+b.
[tex]1 = {27}^{1 + x - 8} [/tex]
4) Simplify 1 + x - 8 to x - 7.
[tex]1 = {27}^{x - 7} [/tex]
5) Use Definition of Common Logarithm: b^a = x if and only if logb (x) = a.
[tex] log_{27}1 = x - 7 [/tex]
6) Use Change of Base Rule.
[tex] \frac{ log_1 }{ log{27} } = x - 7[/tex]
7) Use rule of 1: log 1 = 0.
[tex] \frac{0}{ log_{27}} = x - 7[/tex]
8) Simplify 0/log_27 to 0.
[tex]0 = x - 7[/tex]
9) Add 7 to both sides.
[tex]7 = x[/tex]
10) Switch sides.
[tex]x = 7[/tex]
Therefor, the answer is x = 7.
