we are given
[tex] 2^p=5^k=10^r [/tex]
Let's assume a term
[tex] 4^p [/tex]
we can write as
[tex] 4^p= (2*2)^p [/tex]
[tex] 4^p= 2^p*2^p [/tex]
now, we know that
[tex] 2^p=5^k=10^r [/tex]
so, we can replace 2^p
[tex] 4^p= 5^k *10^r [/tex]
now, we can solve for p
we can take ln on both sides
[tex] ln(4^p)= ln(5^k *10^r) [/tex]
now, we can use property of log
[tex] p*ln(4)= ln(5^k)+ln(10^r) [/tex]
we can simplify it further
[tex] p*ln(4)= kln(5)+rln(10) [/tex]
we get
[tex] p=\frac{kln(5)+rln(10)}{ln(4)} [/tex]................Answer
