Solve for x: Log(base 3)(X) + log(base 3)(x+4) = 8

Question

There are two potential roots, A and B, where 

A = ?

B = ?

Is A actually a root? (yes or no)

Is Bactually a root? (yes or no)

Answer 1

log₃ x + log₃ (x + 4) = 8

log₃ x(x + 4) = 8

3⁸ = x² + 4x

6561 = x² + 4x

0 = x² + 4x - 6561

Use the Quadratic Formula to determine x. Remember a logrithm cannot be taken of a number less than 0.

Answer 2

Answer 3

x ^2 +4x = 3^8= 6561

x =-4/2 +/-(1/2)sqr(16+4(6561)=-2+/-sqr6565

x=-2 +/-sqt6565= about 79.02 and -83.02

A<0 and not a real root

B=-2+sqr6565 is a real root

generally it's said logs can't be taken of negative numbers

but using Euler's identity, imaginary roots are possible

e^ix = cosx +isinx

let x= pi

then e^ipi =-1

and

ipi=ln(-1)= the natural log of a negative number

= an imaginary number times a transcendental number

Atheist philosopher/mathematician Bertrand Russell (not to be confused with the Jack Russell canine) called Eurler's identity the most beautiful equation in mathematics, Euler was perhaps the most prolific writer of mathematics.

Legend had it Czarina Catherine the Great invited Euler and Diderot to debate at her Russian court. She liked to amuse herself watching intellectuals duke it out. Euler for the first time proved his Idenity. Then said "QED. Therefore God Exists." Atheist philosopher Denis Diderot had no response and left the court in disgrace, traveled back to France dumbfounded. .

There's an episode of the Big Bang Theory, where they bet whether Raj can hold his breath until their Euler disc stops moving, it's the same Euler. The disc is named after him.