Exponentials & Logarithms

Exponential functions

Functions of the form:

f(x) = a^x

are known as exponential functions. The graphs of all such exponential functions pass through (0, 1).

Logarithms

Logarithms are another way of writing indices.

• If a = b^c then c = log_ba

Example

We know that 10² = 100
Therefore, log₁₀100 = 2

You may often see ln x and log x written, with no base indicated. It is generally recognised that this is shorthand:

• log_ex = lnx

• log₁₀x = lgx or logx (on calculators)

Remember that e is the exponential function, equal to 2.71828…

Laws of Logs

The properties of indices can be used to show that the following rules for logarithms hold:

• log_ax + log_ay = log_a(xy)

• log_ax – log_ay = log_a(x/y)

• log_axⁿ = nlog_ax

Example

Simplify: log 2 + 2log 3 - log 6
= log 2 + log 3² - log 6
= log 2 + log 9 - log 6
= log (2 × 9) - log 6
= log 18 - log 6
= log (18/6)
= log 3

NB: In the above example, I have not written what base each of the logarithms is to. This is because for the laws of logarithms, it doesn't matter what the base is, as long as all of the logs are to the same base.

Another important law of logs is as follows. This is a very useful way of changing the base (in this formula, the base does matter!). Most calculators can only work out ln x and log₁₀x (usually just written as 'log' on the button) so this formula can be very useful.

Example

Calculate, to 3s.f., log₃5

log₃5 = log₁₀5 = 1.46 (3s.f.)
           log₁₀3

Solving Equations

Logarithms can be used to help solve equations of the form a^x = b by "taking logs of both sides".

Example

Solve 2^x = 6

Then log(2^x) = log(6) [we are allowed to take logs of both sides like this]

x log(2) = log(6) [using one of the 'laws of logs']

x = log(6)
      log(2)

(= 2.58...)

Notice how I haven't said what the base is. This is because it doesn't matter, as long as they are both the same.