Indices and laws of logarithms

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Transcript of Indices and laws of logarithms

  1. 1. 46: Indices and Laws of46: Indices and Laws of LogarithmsLogarithms Christine Crisp Teach A Level MathsTeach A Level Maths Vol. 1: AS Core ModulesVol. 1: AS Core Modules
  2. 2. Indices and Laws of Logarithms Module C2 "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"
  3. 3. Indices and Laws of Logarithms Unknown Indices Because of important practical applications of growth and decay, we need to be able to solve equations of the type ba x = where a and b are constants. Equations with unknown indices are solved using logarithms. We will see what a logarithm is and develop some rules that help us to solve equations. We have met the graph of and seen that it represents growth or decay. x ay =
  4. 4. Indices and Laws of Logarithms Ans: If we notice that 3 101000 = e.g. How would you solve ?100010 =x
  5. 5. Indices and Laws of Logarithms e.g. How would you solve Ans: If we notice that 3 101000 = 3= x 3 1010 =x We can use the same method to solve 813 =x or 2552 =x 122 == xx4= x 4 33 = x 22 55 = x then, (1) becomes - - - - (1)100010 =x
  6. 6. Indices and Laws of Logarithms We need to write 75 as a power ( or index ) of 10. Suppose we want to solve 7510 =x This index is called a logarithm ( or log ) and 10 is the base. Our calculators give us the value of the logarithm of 75 with a base of 10. 8751 1010 x The value is ( 3 d.p. ) so,8751 8751 x Tip: Its useful to notice that, since 75 lies between 10 and 100 ( or ), x lies between 1 and 2. 21 1010 and The button is marked log
  7. 7. Indices and Laws of Logarithms A logarithm is just an index. To solve an equation where the index is unknown, we can use logarithms. e.g. Solve the equation giving the answer correct to 3 significant figures. 410 =x x is the logarithm of 4 with a base of 10 4log410 10== xx We write In general if bx =10 then bx 10log= = logindex ( 3 s.f. )6020=
  8. 8. Indices and Laws of Logarithms ( 2 d.p. )362= x Solution: 230log 10=x (b) 50log2 10 =x 30102 = x 150= x ( 2 d.p. ) Exercise 23010 =x Solve the following equations giving the answers correct to 2 d.p. (a) (b) 50102 =x (a) )32(