How to solve the equation $\log_3{x}+\log_3{\sqrt{x}}+\log_3{x^{1/4}}+...=4$

Question

$$\log_3{x}+\log_3{\sqrt{x}}+\log_3{x^{1/4}}+...=4$$

How to solve the following equation?

Answer 1

Hint: Use the laws of logarithms. For example $\log_3x^{1/4}=\frac 14\log_3x$ This will give you a geometric series to sum.

Answer 2

Another method, sum the logarithms to get: $$\log_3 \left(x\prod_{i=1}^\infty x^{1/2i} \right)=\log_3 \left(x^{1+1/2+1/4+\ldots} \right)=2\log_3 x =4$$

Answer 3

1) $\log_n a + \log_n b = \log_n ab$

2) $x^ax^b = x^{a+b}$

3) $1 + \frac 12 + \frac 14 + .... = ???$

4) $\log_3 K = 4 \implies ?what?$