$8^a=3$ and $3^b=5$ and $10^c=5$ then find $c$ in terms of $a$ and $b$.

Question

if $8^a=3$ and $3^b=5$ and $10^c=5$ then find $c$ using $a$ and $b$.

My Attempt:

if $8^a=3$ and $3^b=5$ then we can say that $8^{ab}=5$ and then we have $2^{3ab}=10^c$ but i cant solve this equation.

Answer 1

We have $$2^{3a}=3\implies 2^{3ab}=5\implies 2\times 2^{3ab}=10\implies 2^{3ab+1}=10$$

Thus $$10^c=2^{c(3ab+1)}$$ But we know that $2^{3ab}=5$ so we are asked to solve $$c(3ab+1)=3ab\implies c=\frac {3ab}{3ab+1}$$

Answer 2

$$8^a=3\implies3a\log2=\log3$$

Similarly, $$b\log3=\log5$$ and $$\log5=c(\log5+\log2)\iff(c-1)\log5=c\log2$$

$$\implies3a\log2\cdot b\log3\cdot(c-1)\log5=\log3\cdot \log5\cdot c\log2$$

$$\iff3a\cdot b(c-1)=c$$

See: Laws of Logarithms