express thee product of: log3 [4th root] of x?

Question

the 3 after log is a lowered 3. Please show work and explain how it was accomplished so I will know how to do this in the future.

Answer 1

log3(x^1/4) = 1/4 * log3(x)
= 1/4 * log10(x) / log10(3)

Answer 2

Since the 4th root of x can be written using a 1/4 exponent, we first rewrite it like that:

log3 x^(1/4)

There is a property of logs that says logb X^m = m*logb X, so we apply that rule here:

(1/4)*log3 x

Answer 3

Law of Logrithms
log a^(b) = b.log a

In the following let log read as log base 3
log x^(1/4) = (1/4).log x

This law may be verified by considering a log base 10 expression:-
log (100) = 2
log (10²) = 2 log 10 = 2 x 1 = 2 (as above)

Answer 4

This is not a product. it is a base 3 logarithm evaluated at the fourth root of x for some number x.

It can be written as the base 3 logarithm of x raised to the 1/4 power.

(base 3) log x^(1/4)

The power rule of logarithms: log x^m = m(log x) for any base.

(base 3) (1/4)(log x)

This is "one-fourth the base 3 logarithm of x".