>>546
2^x＝5^y＝10^z = kとして
x*log2 = logk
y*log5 = logk
z*log10 = logk

x = logk/log2
y = logk/log5
z = logk/kog10

1/x + 1/y = log2/logk + log5/logk = log10/logk = 1/z