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Exponential functions y=a x

Exponential functions y=a x. What do they look like ? y= 2 x looks like this. Y=2 x. Y=10 x looks like this. Y=10 x. Y=2 x. Y=3 x looks like this. Y=10 x. Y=3 x. Y=2 x. y=e x looks like this. y=3 x. “e” is a special number in maths, It’s value is 2.718281828.

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Exponential functions y=a x

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  1. Exponential functions y=ax What do they look like ? y= 2xlooks like this

  2. Y=2x

  3. Y=10x looks like this Y=10x Y=2x

  4. Y=3x looks like this Y=10x Y=3x Y=2x

  5. y=ex looks like this y=3x “e” is a special number in maths, It’s value is 2.718281828. We will explain the importance of the number e in a later lesson!! y=ex y=10x y=2x

  6. All these exponential functions have inverses To find INVERSE We reflect the function in the line y=x

  7. y=10xand y=ex are the most important y=ex y=10x The inverse functions are called Logarithms y=ln(x) y=log(x)

  8. In General for y=ax Remember ff-1(x) = f-1f(x) = x Log10(x) is written as simply Log(x) Loge(x) is written as Ln(x)  Natural or Naperian Log

  9. So what ? Logarithms allow us to solve equations involving exponentials like : 10X=4 where x is the power Take logs of both sides Because we are taking ff-1(x) FUNCTION ax (EXPONENTIAL) INVERSE FUNCTION (LOG)

  10. So if 10x=4 then x=Log(4) The power “x” is therefore a logarithm !! Logarithms are powers in disguise !! And so the laws of logs are a little like the laws of indices

  11. Log Laws – Rule 1 Indices Logs Log Laws – Rule 2 Indices Logs

  12. Log Laws – Rule 3 Rise both sides to power a Why? LHS ff-1(x)=x This is perhaps the most useful Rule Use the laws of indices on RHS RHS ff-1(x)=x

  13. Log Laws – Rule 4 This equals a1 Because ff-1(x) Why? Log Laws – Rule 5 All logs pass through (1,0)

  14. Log laws - Rule 6 Using law 2 SO because Loga1=0

  15. Log laws - Rule 7 Why? The change of base rule Take Logs of both sides Using Log Law 3 BUT y=logab

  16. All together

  17. What now 1- The laws of logarithms are given to you in an exam, you don’t have to remember them 2- But you do have to use them 3- We use logarithms to solve things like ax=b 4- And now you know why!! Because they undo the exponential ax ; as they are it’s Inverse : Next we will use logarithms

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