Exponents & Logarithms (DP IB Maths: AI SL)

What are the laws of indices?

• Laws of indices (or index laws) allow you to simplify and manipulate expressions involving exponents
  • An exponent is a power that a number (called the base) is raised to
  • Laws of indices can be used when the numbers are written with the same base
• The index laws you need to know are:
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• These laws are not in the formula booklet so you must remember them

How are laws of indices used?

• You will need to be able to carry out multiple calculations with the laws of indices
  • Take your time and apply each law individually
  • Work with numbers first and then with algebra
• Index laws only work with terms that have the same base, make sure you change the base of the term before using any of the index laws
  • Changing the base means rewriting the number as an exponent with the base you need
  • For example, 
  • Using the above can them help with problems like 

What are logarithms?

• A logarithm is the inverse of an exponent
  • If then  where a > 0, b > 0, a ≠ 1
    • This is in the formula booklet
    • The number a is called the base of the logarithm
    • Your GDC will be able to use this function to solve equations involving exponents

• Try to get used to ‘reading’ logarithm statements to yourself
  • would be read as “the power that you raise to, to get , is ”
  • So  would be read as “the power that you raise 5 to, to get 125, is 3”

• Two important cases are:
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    • Where e is the mathematical constant 2.718…
    • This is called the natural logarithm and will have its own button on your GDC
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    • Logarithms of base 10 are used often and so abbreviated to log x

Why use logarithms?

• Logarithms allow us to solve equations where the exponent is the unknown value
  • We can solve some of these by inspection
    • For example, for the equation 2^x = 8 we know that x must be 3
  • Logarithms allow use to solve more complicated problems
    • For example, the equation 2^x = 10 does not have a clear answer
    • Instead, we can use our GDCs to find the value of