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Exponentials and Logs

Exponentials and Logs . Modules 12, 13, 14, 15 October 23, 2012. Inverse Functions. Logs and exponentials are inverses of each other and can be rewritten in this way: We can use the opposite function to isolate our variable when we solve equations.  . What is an exponential function?.

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Exponentials and Logs

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  1. Exponentials and Logs Modules 12, 13, 14, 15 October 23, 2012

  2. Inverse Functions • Logs and exponentials are inverses of each other and can be rewritten in this way: • We can use the opposite function to isolate our variable when we solve equations. 

  3. What is an exponential function? • Exponential functions are of the form: • Our variable here is still x. • Ex. where a > 0 and a ≠ 1.

  4. What is a log function? • Log functions are of the form: • “What power do I raise the base a to in order to get the argument x?” • Ex. where a > 0 and a ≠ 1.

  5. Transformations • Exponential and log functions can also have transformations just like the functions did from the first exam material. • Ex.

  6. Is this a valid exponential function?

  7. Is this a valid exponential function? • No, because here the base value would have to be a = -3, and we know that a has to be positive.

  8. Is this a valid exponential function?

  9. Is this a valid exponential function? • Yes, because our base is a=3, which is valid. The negative out front is a reflection over the x-axis because it’s not being raised to the x power.

  10. Is this a valid exponential function?

  11. Is this a valid exponential function? • Yes, because our base is a=2/3, which is valid because fractions are okay.

  12. Is this a valid log function?

  13. Is this a valid log function? • Yes, because our base is a=π, which is valid because it is a value positive and not equal to 1. The 2 in front of the x is a horizontal transformation, which causes the graph to compress horizontally by ½.

  14. Is this a valid log function?

  15. Is this a valid log function? • Yes, because our base is a=7, which is valid because it is a value positive and not equal to 1. The 4 can come out front by log rules, and so it will end up vertically stretching by 4.

  16. Is this a valid log function?

  17. Is this a valid log function? • No, because our base is a = -2, and negative numbers aren’t allowed.

  18. Log Properties: • Identity: • Inverse (I): • Inverse (II): • Exponent to Constant: • Product: • Quotient:

  19. Graphing • Both types of function act differently when a is between 0 and 1 than they do when a is above 1. • Think about what happens when we square a number: •  gets smaller •  gets bigger • So you need to memorize 4 basic graph shapes.

  20. Graphing Notes • Also think about the asymptotes to help you think about where they end up when you transform the graphs. • AND… • Our good old friends domain and range.

  21. Domain: all reals Range: (0, infinity) Domain: all reals Range: (0, infinity) Domain: (0, infinity) Range: all reals Domain: (0, infinity) Range: all reals

  22. Graph a function: • First we have a base of a=2, so that tells us we need to start with the graph of . Then we’ll reflect it over the x axis to turn it into , before we shift the whole thing up by 3 units to get to our final answer of .

  23. Graph a function: • First we have a base of a=2, so that tells us we need to start with the graph of . Then we’ll reflect it over the x axis to turn it into , before we shift the whole thing up by 3 units to get to our final answer of . New asymptote will be horizontal and at y=3 because it started at y=0 and there was a 3 unit vertical change. Domain: all reals Range: (- infinity, 3)

  24. Solving Equations • Remember that logs and exponentials are inverse functions and we can use them to “undo” each other. Sometimes we need to do this, sometimes we don’t. There are usually a couple ways to solve. • For example:

  25. Solving Equations •  realize that 4 = 2^2 •  when we have the same base, we can set the arguments equal to each other

  26. Solving Equations  When we have variables in the exponent, we need to take the log of both sides to “get it out” so we can solve for it.  Log rules

  27. Solving Equations  We want to get x by itself, so we need to raise both sides to the x power (or reorganize using the definition).  Take the cube root of both sides.

  28. Expanding Logs • Sometimes we have really complicated logs which we can expand into many individual terms using log rules. • - • -

  29. Condensing Logs • We also want to be able to pull many logs into a single log in other situations. • Note that we cannot combine them unless they have the same base! • + • +

  30. Rewriting Bases • Sometimes we’re in the situation when we need an exact value of a log using our scientific calculator, but the base is something besides e or 10. • We can rewrite any log using this formula so that it’s possible to compute: • Ex. • = 0.7924 • = 2.3219

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