Exponents and Logs

Further information on this topic can be found in Chapter 6 of:

Maths from Scratch for Biologists

Numerical ability is an essential skill for everyone studying the biological sciences but many students are frightened by the 'perceived' difficulty of mathematics, and are nervous about applying mathematical skills in their chosen field of study. Maths from Scratch for Biologists is a highly instructive, informal text that explains step by step how and why you need to tackle maths within the biological sciences. (Amazon.co.UK)

Before we begin:

An "index" or exponent of a number indicates how many times the number or term (the "base") should be multiplied by itself.

Multiplication is a shortcut for addition:

Exponents are a shortcut for multiplication:

Similarly, 

and

As you will see, logarithms are a shortcut for exponents.

Exponential Functions:

Any quantity which increases by being multiplied by the same value at regular intervals is said to grow "exponentially".
Exponential functions occur frequently in biology because they describe processes of growth and decay, for example:

• The Beer-Lambert Law:   As light is transmitted through a substance of thickness x, the change in its intensity (absorbance) is defined by:

I(x) = I₀e^-kx

where I(x) is the light intensity at any given thickness of the absorber, I₀ is the initial (incident) light intensity and k is a specific constant for any particular substance.
The value of this in biology lies in analytical spectroscopy. An unknown concentration of a substance can be determined by measuring the amount of light that a sample absorbs and applying Beer's law. If the absorption (or "extinction") coefficient is not known, the concentration of the substance can be determined using a graph of absorbance versus concentration derived from standard solutions of known concentration.

• Radioactive decay:

N = N₀e^-λt

where N is the amount of radioactivity remaining at time t, N₀ the original amount of radioactivity and λ the decay constant for the radionuclide.

• Bacterial growth: (see below)

N(t) = N₀2^t/T₀

Logarithms

"Logarithm" comes from the Greek logos, "to calculate" and arithmos, "a number". In other words, a logarithm is a "calculating number".

A logarithm is: the exponent or power to which a base must be raised to yield a given number:

Logarithms are written as the sum of an integer (the "characteristic") and a decimal (the "mantissa") (e.g. log₁₀ 150 = 2.176).

• To multiply two numbers, add their logs.

• To divide two numbers, subtract their logs.

Logs are also useful for many other calculations, e.g. to work out the fifth root of a number, divide the log of the number by 5:
The fifth root of 10 000 000 000 = 10/5 = 2 = 100

Logs can be converted back to real numbers by looking up tables of antilogarithms, the inverse of the log function, or on a calculator as follows:

• the key with x = log value

• the key then the log key

• the key with y = 10 and x = log value

Most commonly encountered logs (common logs) are in base 10, written as log₁₀.

Logs to the base "e" (2.71828...) are known as natural logs and are written as ln.

e is Euler's number. Leonard Euler was a Swiss mathematician who described e in his manuscript, Meditatio in Experimenta Explosione Tormentorum Nuper Instituta (Meditation on Experiments Made Recently on the Firing of Cannon), written in 1728.  e is a very unusual number. Apart from being the base for natural logarithms, it is an irrational (its decimal expansion never terminates, nor is it eventually periodic) and transcendental number (not the root of any polynomial with rational number coefficients) - these are properties that e shares with π.

Rules of Logarithms:

Where a is a positive number that does not equal 1, n is a real number (i.e. all numbers representable by an infinite decimal expansion, i.e. the ratios a/b, where a and b are integers and b does not equal 0) and x and y are positive real numbers:

Rule 1:   log_a (x.y) = log_a (x) + log_a (y)

Rule 2:   log_a (x/y) = log_a (x) - log_a (y)

Rule 3:   log_a (x)ⁿ = n log_a (x)

So why bother?

Just as exponents are a shortcut for multiplication, logs are a shortcut for exponents.

Logs can be used to solve exponential equations, which occur commonly in biology, e.g:

pH = -log [H⁺]

pH = -log 4*10^-8 = 7.4

• The exponential growth of populations:

The value of µ can be determined from the following equation:

the natural log of the number of cells at time t minus the natural log of the number of cells at time zero (t₀) equals the growth rate constant multiplied by the time interval.
For most purposes, it is easier to use log₁₀ values rather than natural logs, so the above equation can be converted as follows:

or alternatively:

Example:

By measuring the increase in the number of cells during a certain time period, the growth rate constant (µ) can be calculated:

t0 = 1.5h: N = 8.4*101, log10 N = 1.92 t = 8.5h: N = 3.39*108, log10 N = 8.53

Therefore in this case:

• Decibel (dB) scale of sound intensity:

dB = 10 log₁₀ (I/I₀₎

where I is the intensity of the sound and I₀ is the reference sound intensity (1*10^-12 W m^-2 , the lowest sound intensity detectable by the human ear):

Why do we use decibels? The ear is capable of hearing a very large range of sound - more than a billion-fold difference in intensity. To deal with such a range, logarithmic units are most useful.

Examples:

The sound intensity at a heavy metal concert is recorded as 0.5*10^-1 W m^-2. What is the intensity of the sound in decibels?

dB = 10 log₁₀ (I/I₀)

= 10 * log₁₀ (0.5*10^-1 / 1*10^-12)

= 10 * 10.7

= 107 dB

The engine of a Ferrari makes a noise of 105 dB and a Ford engine makes a noise of 99 dB. How many times more intense is the Ferrari engine noise?

Difference in noise = 105 - 99 = 6 dB

Remember that a difference of 10 dB represents a 10-fold diffence in sound intensity, and a difference of 3 dB represents roughly a doubling in the intensity of sound, so the Ferrari is four times louder than the Ford.

© MicrobiologyBytes 2005.