Given that N > 1 and
\(\dfrac{1}{\log_2 N} + \dfrac{1}{\log_4 N} + \dfrac{1}{\log_6 N} + \dfrac{1}{\log_8 N} + \dfrac{1}{\log_{10} N} = \dfrac{1}{\log_x N}\)
find the value of the integer x.
Solve for x:
log(2)/log(N) + log(4)/log(N) + log(6)/log(N) + log(8)/log(N) + log(10)/log(N) = log(x)/log(N)
log(2)/log(N) + log(4)/log(N) + log(6)/log(N) + log(8)/log(N) + log(10)/log(N) = log(x)/log(N) is equivalent to log(x)/log(N) = log(2)/log(N) + log(4)/log(N) + log(6)/log(N) + log(8)/log(N) + log(10)/log(N):
log(x)/log(N) = log(2)/log(N) + log(4)/log(N) + log(6)/log(N) + log(8)/log(N) + log(10)/log(N)
Multiply both sides by log(N):
log(x) = log(2) + log(4) + log(6) + log(8) + log(10)
log(2) + log(4) + log(6) + log(8) + log(10) = log(2 4 6 8 10) = log(3840):
log(x) = log(3840)
Cancel logarithms by taking exp of both sides:
x = 3840