Question

25. The value of x in the following:
3^X+1 = 1/27​

Answer 1

Answer:

The real value of x =2 satisfies the equation, using logs as others have shown.

However, there are an infinity of OTHER solutions to the equation.

They all satisfy:

x = 2 ( 1 + i * pi * n /log(3))

(which is the correct answer) where:

i is the square root of -1, so i * i = -1

pi = the ratio of the length of the circumference of a circle to its diameter, approximately 3.1415927

log = the natural logarithm function, i.e. log to the base e, where e is approximately 2.718283

n = any integer, positive or negative

Put n = 0 for the real solution:

x = 2 * (1 + 0) = 2

All other values of n i.e. …. -3, -2, -1, 1, 2, 3 … correspond to an infinity of distinct complex roots of 3^(x+1) = 27

For example, for n = 1

x = 2 ( 1 + i * pi / log(3)), which is approximately 2 + 5.7192 * i

i.e. 3^(2 + 5.7192 * i + 1) = 3^(3 + 5.7192 * i) =~ 27