2^a=5^b=10^c...can you help me solve this..?

Question

the answer is a=(bc)/(b-c)...but can you show me how to get it

Answer 1

2^a=5^b=10^c

5^b=10^c
5^b=2^c*5^c now divide both sides by 5^c
5^b/5^c=2^c
5^(b-c)=2^c now multiply both sides by 2^(b-c)
5^(b-c)*2^(b-c)=2^c*2^(b-c)
10^(b-c)=2^b now take the (b-c) root of both sides
10=2^(b/(b-c)) now raise both sides to the c power
10^c = 2^(b/(b-c))^c
10^c=2^(bc/(b-c))
We already know that 2^a = 10^c, so
2^a=2^(bc/(b-c))
If the roots are the same and the expressions are equal, then the exponents are equal, so
a=bc/(b-c)