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Bell work

Bell work. Find the value to make the sentence true. NO CALCULATOR!! 2 3 = x 5 x = 25 (½) 3 =x X (1/2) = 5. Graph f(x) = 2 x Name 3 points on f(x) Graph y = x Graph f -1 (x) Write an equation for the inverse.

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Bell work

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  1. Bell work Find the value to make the sentence true. NO CALCULATOR!! • 23 = x • 5x = 25 • (½)3=x • X (1/2) = 5

  2. Graph f(x) = 2x • Name 3 points on f(x) • Graph y = x • Graph f-1 (x) • Write an equation for the inverse.

  3. F(x) = 2xto write the equation of the inverse, switch the x and y values x = 2y How do you solve for y? Verbally we would say that “Y is the exponent that 2 to raised to in order to get x.” To write it mathematically: y = log2x

  4. A logarithm is just an exponent • So the inverse of an exponential function is a logarithmic function. (This means that exponentiation “undoes” logarizing and vice versa.) Definition of logarithm: X = ay can be rewritten as logax = y (a>0, a≠1, x>0) ** Why is domain positive? ** What # can be raised to power and give you a negative? **

  5. Any exponential expression can be written as a logarithmic expression and vice versa Rewrite in exponential form. • log28 = 3 • log381=4 • log164 = ½ • log273 = ⅓

  6. Any exponential expression can be written as a logarithmic expression and vice versa Rewrite as a logarithmic expression. • 52 = 25 • 34 = 81 • (½)3 = ⅛ • (2)-2 = ¼

  7. Remember: A logarithm is an exponent Evaluate. • log216= ** Think: 2 to what power gives me 16?** • log327= _____ • log22=_____ • log101000=____ • 2 log31= _____

  8. Common logarithm • The common logarithm is a log with base of 10. • When the base is 10, we don’t write it!! • Example: log 100 = 2 (Understood base 10) • The calculator uses the common base of 10 when you plug in a value. • Use the calculator to find log 10

  9. Natural logarithm • The natural logarithm is a log with base e. • Abbreviation is ln • loge5 will be written as ln 5 * “ln” means the base is understood to be e* What is the value of ln e?

  10. Some other important properties that always hold true: • loga1=0 • logaa=1 • logaax=x • ln1 = 0 • ln e = 1 • lnex=x

  11. Simplify. • log55x • 7log714 • log51

  12. If logax=logay, then x=y • Solve for x: log2x=log23 • Solve for x: log44= x

  13. Steps to graphing a logarithmic function • Rewrite as an exponential equation • Pick values for Y and solve for x • Plot the points.

  14. Graph y = log 2x • Rewrite as Exponential equation: ___________ • Domain: • Range: • Intercepts: • Asymptotes: • How is this related to the graph y =2x?

  15. Graph f(x) = log3x • Rewrite as an exponential equation: ___________________ • Domain: • Range: • Intercepts: • Asymptotes:

  16. Graph y = log 4 x • Rewrite as an exponential equation: ___________________ • Domain • Range • Intercepts • Asymtpotes

  17. Graph y = lnx • Rewrite as an exponential equation: ___________________ • Domain: • Range: • Intercepts: • Asymptotes:

  18. Graph y = log3(x-2) • Rewrite as an exponential equation: ___________________ • Domain: • Range: • Intercepts: • Asymptotes:

  19. Transformations • What happens to the graph of f(x±c)? Moves right or left c units • What happens to the graph of f(x) ± c? Moves the graph up or down c units • What happens to the graph of f(-x)? Reflects across the y axis • What happens to the graph of –f(x)? Reflects across the x axis

  20. Suppose f(x) = log2xDescribe the change in the graph • G(x) = log2(-x) Reflect over the y axis • G(x) = log2 (x+5) Moves 5 units to the left, VA: x = -5 • G(x) = -log2(x) Reflect over the x axis • G(x) = log2x-4 Moves the graph down 4 units • g(x) = log 2 (x-3) Moves the graph 3 units to the right, VA: x = 3

  21. Sketch the following graphsDon’t forget RXSRY • Y = -lnx 2. Y = ln(-x) • Y = ln(x-2) • Y = lnx + 3

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