C2: Logarithms

1. C2: Logarithms Learning Objective: to be able to write an expression in logarithmic form

2. 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)

3. Logarithms Find p if p3 = 343. We can solve this equation by finding the cube root of 343: 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.

4. Logarithms This is defined as: y = logax ay= x The expressions and are interchangeable. y = logax To avoid using trial and improvement we need to define the poweryto which a given base a must be raised to equal a given number 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 For example, 25 = 32 can be written in logarithmic form as: log2 32 = 5

5. Logarithms y = loga x ay = x We have that: So: y = logaay Also: and Taking a log and raising to a power are inverse operations. For example: 2 6

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)

7. Task 1 : • Exercise 3B & 3C