Given 2^p=5^k=10^r , express p in terms of k and r

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