Expand log 4 √ x

Answer 1

y = log 4 √x

Assuming this means logarithm base 4 of the square root of x.

y = log_4 (SQRT(x))

Roots are just fractional powers.

y = log_4 (x)^(1/2)

Powers or exponents of logs can be brought forward.

y = (1/2) * log_4 (x)

All logs can be rewritten in exponent form.

4^y = x^(1/2)

Swap sides

x^(1/2) = 4^y

Square both sides

x = (4^y)^2

A power to a power multiplies.

x = 4^2y

Rewrite

x = (4^2)^y

Simplify

x = 16^y

Equivalent log form

y = log_16 (x)

Change of base:

y = ln (x) / ln (16)

The natural logarithm expands into a Taylor Series:

ln(x) = (x-1) - (x-1)^2 / 2 + (x-1)^3 / 3 - (x+1)^4 / 4 + ...

y = log_4 √x = ln (x) / ln (16)

ln(x) = (x-1) / ln (16) - (x-1)^2 / 2*ln (16) + (x-1)^3 / 3*ln (16) - (x+1)^4 / 4*ln (16) + ...

In compact form:

y = log_4 √x = ∑ (from n=1 to infinity) of ((-1)^(n-1) * x^n] / [n *ln(16)] for |x| < 1