Solution:

Given equation: log2x log4x log16x=21/4

To solve this equation for x, we will use the properties of logarithms and simplify the equation.

Step 1: Simplify the logarithms

We know that loga + logb = log(ab). Using this property, we can simplify the first two logarithms in the equation.

log2x + log4x = log(2x * 4x) = log(8x^2)

Now, we can rewrite the equation as:

log(8x^2) log16x = 21/4

Step 2: Simplify the logarithms further

We know that loga - logb = log(a/b). Using this property, we can simplify the left-hand side of the equation.

log(8x^2/16x) = log(x/2)

Now, the equation becomes:

log(x/2) = 21/4

Step 3: Solve for x

We know that loga(b) = c is equivalent to b = a^c. Using this property, we can solve for x.

x/2 = 2^(21/4)

x = 2^(21/4 + 1)

x = 2^(25/4)

x = 32

Therefore, x is equal to 32.

Final Answer: x = 32

Log 2 X + Log 2^2 X + Log 2^4 X =21/4
1log 2 X + 1/2Log 2 X + 1/4Log 2 X =21/4
lets take a common (Log 2 X) then.
(1+1/2+1/4)Log 2 X = 21/4
7/4 Log 2 X = 21/4
log 2 X = 21/4 × 4/7
Log 2 X = 3
(2)^3 =8
Ans is 8....