C2: Logarithms

Learning Objective: to be able to write an expression in logarithmic

form

Logarithmic functions are the inverses of the exponential functions.

The graph of a logarithmic function is the inverse of its exponential function (ie a reflection in the line y=x)

LogarithmsFind p if p3 = 343.

We can solve this equation by finding the cube root of 343:3= 343p

= 7p

Now, consider the following equation:

Find q if 3q = 343.

We need to find the power of 3 that gives 343.

One way to tackle this is by trial and improvement.

Use the xy key on your calculator to find q to 2 decimal places.

LogarithmsTo avoid using trial and improvement we need to define the power y to which a given base a must be raised to equal a given number x.

This is defined as: y = loga x

“y is equal to the logarithm, to the base a, of x”

This can be written using the implication sign :

y = loga x ay = x y = loga x ay = x

The expressions y = loga x ay = xand are interchangeable.

For example, 25 = 32 can be written in logarithmic form as:

log2 32 = 5

LogarithmsTaking a log and raising to a power are inverse operations.

We have that: y = loga x ay = x y = loga x ay = x

So: log =a xa x

Also: y = loga ayy = loga ay

For example:

7log 27 = 2 and 63log 3 = 6

Examples:

• Rewrite as a logarithm 54 = 625

• 54 = 625

• 4 = log5 625

• Find the value of log3 81

• log3 81 = x

• 3x =81

• x = 4 (because 34 =81)

Task 1 :

• Exercise 3B & 3C