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Exponents & Logarithms (DP IB Applications & Interpretation (AI)) : Revision Note

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Laws of Indices

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:

    • left parenthesis x y right parenthesis to the power of m equals x to the power of m end exponent y to the power of m

    • stretchy left parenthesis x over y stretchy right parenthesis to the power of m equals x to the power of m over y to the power of m

    • x to the power of m cross times x to the power of n equals x to the power of m plus n end exponent

    • x to the power of m divided by x to the power of n equals x to the power of m minus n end exponent

    • stretchy left parenthesis x to the power of m stretchy right parenthesis to the power of n equals x to the power of m n end exponent

    • x to the power of 1 equals x

    • x to the power of 0 equals 1

    • 1 over x to the power of m equals x to the power of negative m end exponent

  • 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, 9 to the power of 4 equals left parenthesis 3 squared right parenthesis to the power of 4 equals 3 to the power of 2 cross times 4 end exponent equals 3 to the power of 8

    • Using the above can them help with problems like 9 to the power of 4 divided by 3 to the power of 7 equals 3 to the power of 8 divided by 3 to the power of 7 equals 3 to the power of 1 equals 3

Examiner Tips and Tricks

  • Index laws are rarely a question on their own in the exam but are often needed to help you solve other problems, especially when working with logarithms or polynomials

  • Look out for times when the laws of indices can be applied to help you solve a problem algebraically 

Worked Example

Simplify the following equations:

i) fraction numerator left parenthesis 3 x squared right parenthesis left parenthesis 2 x cubed y squared right parenthesis over denominator left parenthesis 6 x squared y right parenthesis end fraction.

 

ai-sl-1-1-2-laws-of-indices-we-i

ii) left parenthesis 4 x squared y to the power of negative 4 end exponent right parenthesis cubed left parenthesis 2 x cubed y to the power of negative 1 end exponent right parenthesis to the power of negative 2 end exponent.

 

ai-sl-1-1-2-laws-of-indices-we-ii


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Introduction to Logarithms

What are logarithms?

  • A logarithm is the inverse of an exponent

    • If a to the power of x equals b then log subscript a open parentheses b close parentheses equals x 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

    • log subscript a left parenthesis b right parenthesis space equals space x would be read as “the power that you raise a to, to get b, is x”

    • So log subscript 5 125 space equals space 3 would be read as “the power that you raise 5 to, to get 125, is 3”

  • Two important cases are:

    • ln space x equals log subscript straight e open parentheses x close parentheses

      • Where e is the mathematical constant 2.718…

      • This is called the natural logarithm and will have its own button on your GDC

    • log space x equals log subscript 10 open parentheses x close parentheses

      • 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 2x = 8 we know that x must be 3

    • Logarithms allow use to solve more complicated problems

      • For example, the equation 2x = 10 does not have a clear answer

      • Instead, we can use our GDCs to find the value of log subscript 2 10

Examiner Tips and Tricks

  • Before going into the exam, make sure you are completely familiar with your GDC and know how to use its logarithm functions

Worked Example

Solve the following equations:

i)

x equals log subscript 3 27,

 

ai-sl-1-1-2intro-to-logs-we-i

ii)

2 to the power of x equals 21.4, giving your answer to 3 s.f.

 

ai-sl-1-1-2intro-to-logs-we-ii
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