Understanding logarithms and natural logarithms in chemistry Crunch Chemistry

A logarithm is simply the number of times we need to multiply one number together to make another number i.e. it is the power to which a number must be raised to get another number. Logarithms allow us to deal with extremely large or extremely small numbers without getting our heads in a spin.

The inverse of this function is

i.e. 10^𝑥and log₁₀𝑥 are inverse functions.

Let’s look at an example.

We often use natural logarithms in chemistry where the base is not 10 but e, Euler’s number.

‘e’ is an irrational number just like ∏ (a number that cannot be accurately expressed as a decimal) and is a fundamental constant with a value of 2.7183…

Graphically speaking, y = e^𝑥 describes an exponential curve.

This is exactly the curve we see if we plot a graph of increasing rate of reaction (y-axis) against increasing temperature (𝑥-axis)

or plot a graph of rate of reaction (y-axis) against time (𝑥-axis) for a first order reaction, y = e^-𝑥

or plot a graph of amount of a radio-nucleotide (y-axis) against time (𝑥-axis), also y = e^-𝑥.

The point is that when we look at the equation y = e^𝑥 (or y = e^-𝑥) we should immediately recognise that y is the observed variable, as plotted on the y-axis and 𝑥 is the controlled variable, as plotted on the 𝑥-axis.

𝑥 can be a whole number, a fraction or an algebraic function as we see in the Arrhenius equation, which describes the relationship between the rate constant for a reaction and temperature:

To reverse an exponential function e^𝑥 we need its inverse, which is a natural logarithm, ln.

There are three laws of logarithms which we need to know:

log a + log b = log (ab) or ln a + ln b = ln (ab)
log a – log b = log (a/b) or ln a – ln b = ln (a/b)
log (a)^b = b x log a or ln (a)^b = b x ln a

It is also worth pointing out that the ratio between log₁₀ and ln is 2.303. What does that mean?

Well, if we were to divide the natural logarithm of a number by the log₁₀ of the same number, the answer is 2.303.

ln 60 / log₁₀ 60 = 2.303

ln 10 = 2.303 and e^2.303 = 10

You can check all this by playing around with the e, ln and log₁₀ functions on your calculator!

We can use natural logarithms to simplify equations like the Arrhenius equation in chemistry. This allows us to plot a linear graph and determine the relationship between the variables more easily where the relationship between the two is not directly proportional.