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Understanding Exponential Functions and Their Inverses through Logarithms

This lesson explores exponential functions of the form y = ax, focusing on specific examples like y = 2^x, y = 10^x, and y = e^x. We delve into the significance of the mathematical constant "e" and its applications. Inverses of these functions, known as logarithms, are introduced, highlighting their ability to solve equations involving exponentials. Key logarithmic rules are discussed, emphasizing their similarity to laws of indices. Understanding these concepts is crucial for tackling advanced mathematical challenges effectively.

jorjanna-gomez
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Understanding Exponential Functions and Their Inverses through Logarithms

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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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