would anyone be able to help me with this question?

if x > 1, then for what positive base b is logb(x) is always equal to 3/4 of log10^x?

Example:
Solve for b:

logb(10000) =3
(log(10000))/(log(b)) = 3 ="3/4 of Log 10^4 or10,000"

Take the reciprocal of both sides:
(log(b))/(log(10000)) = 1/3

Multiply both sides by log(10000):
log(b) = (log(10000))/3

(log(10000))/3 = log(10000^(1/3)) = log(10 10^(1/3)):
log(b) = log(10 10^(1/3))

Cancel logarithms by taking exp of both sides:
Answer: | b = 10 10^(1/3) =10,000^1/3 or "a"

Here's a direct derivation:

.

Here's a direct way using the change-of-base method

log_b (x) = (3/4) log₁₀ (x)

log (x) / log (b)  = (3/4) log (x) / log 10        divide both sides by log (x)  and note that log10  = 1

1/ log (b)  = 3/4       which implies that

log (b)  = 4/3    which implies that

10 ^ (4/3)  = b

[10^4] ^(1/3) = b

10000^(1/3) = b = ∛10000