How can the method of using logarithms to solve equations of the form 10^(mx+n)=c be related to solving scientific problems using logarithms in the form a(10^mx)=c?

Question

How can the method of using logarithms to solve equations of the form 10^(mx+n)=c be related to solving scientific problems using logarithms in the form a(10^mx)=c?

Answer 1

In this case, you first divide both sides by a, so that 10^mx = c/a. Take common log of both sides: log(10^mx) = log (c/a). The logarithm of a power property states that log a^b = b log a, therefore we get (mx)log 10 = log (c/a). Now the common log base is 10, and log_b b = 1. Now it's simple algebra: mx = log (ca), and x = (log (c/a)) /m.

Answer 2

Note that 10^(mx+n) = (10^mx)*10^n.

So, if 10^(mx+n) = a(10^mx) = c, it follows that a = 10^n.