How can logarithms be used to solve equations in the form 10^(mx+n)=c?

Question

How can logarithms be used to solve equations in the form 10^(mx+n)=c?

Answer 1

Take common log of both sides: log(10^mx+n) = log c. The logarithm of a power property states that log a^b = b log a, therefore we get (mx+n)log 10 = log c. Now the common log base is 10, and log_b b = 1. Now it's simple algebra: mx = log c - n, and x = (log c - n) /m.

Answer 2

The Power Property of Logarithms is that:

log(a^b) = b*log(a)



Therefore, logarithms can be used to solve equations in that form (10^(mx+n)=c) by changing the (mx+n) term from an exponent to a coefficient. This makes it much easier to work with.

10^(mx+n)=c

log(10^(mx+n))=log(c)

Now bring the (mx+n) term down from exponent over to coefficient, per the Power Property of logs.

(mx+n)*log(10)=log(c)

Note that log(10)=1, so we can remove that term.

mx+n=log(c)

This is how logarithms can be used to solve equations in the form given.

Hope this helps. :)