How can logarithms be used to solve scientific problems that lead equations in the form a(10^mx)=c?

Question

How can logarithms be used to solve scientific problems that lead equations in the form a(10^mx)=c?

Answer 1

The Power Property of Logarithms is that:

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

Therefore, logarithms can be used to solve equations in that form (a*10^(mx)=c) by changing the (mx) term from an exponent to a coefficient. This makes it much easier to solve for the variable m or x.



a*10^(mx)=c

log(a*10^(mx))=log(c)

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

(mx)*log(a*10)=log(c)

(mx)*log(10a)=log(c)

This equation can be manipulated further to solve for any variable, potentially using other properties of logarithms such as product, quotient and root properties.

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

Hope this helps. :)

Answer 2

Hi Sally A.,

That's strange terminology "lead equations in the form" -- but, I'm assuming that you mean, "Assuming an equation in the form of "a*10^mx = c", how may I use logarithms to reduce my data (and solve for a and m)?

So that's simple. Transform that equation by taking the logarithm (base 10) of each side:

log(a*10^mx) = log(c)

log a + mx = log(c)

Graph this as log(c) = mx + log(a)

That is, use the log(c) data as "y" and the x data as "x". You now have data approximating a line with intercept log(a), and slope m. Obtain these values (a and m) using a least-squares regression fit of the data. All calculation spreadsheets (Excel from Microsoft, Calc from Apache OpenOffice, etc.) have built-in functions for this.

Where your judgement comes in on this, is detecting if the data as graphed above do not fit a straight line. This would indicate a different equation should be applied.

-- Cheers, -- Mr. d.