Logarithmic Functions (Day 1)

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Presentation transcript:

Logarithmic Functions (Day 1) Algebra 2: Section 8.4 Logarithmic Functions (Day 1)

Solving for “x” Addition x – 3 = 5 Subtraction 3 + x = 9 Multiplication 1/2x = 4 Division 5x = 25 Power x3 = 27 Roots If “x” is an exponent?

Definition of Logarithm logby is read as “log base b of y”

Rewrite the equations in exponential form. Examples Rewrite the equations in exponential form. Logarithmic Function Exponential Function 1. log39 = 2 2. log81 = 0 3. log5 (1/25) = -2 1. 32 = 9 2. 80 = 1 3. 5-2 = 1/25

Examples Evaluate the expressions. Hint: For logby ask yourself what power of b gives you y? 4. log464 What power of 4 gives you 64? 4x = 64 Answer: 3 5. log20.125 What power of 2 gives you 0.125? 2x = 0.125 Answer: -3 6. log1/4256 What power of ¼ gives you 256? 1/4x = 256 Answer: -4 7. log322 What power of 32 gives you 2? 32x = 2 Answer: 1/5

Common and Natural Logs Common Logarithm (the base of 10 is not written) log10x = log x Natural Logarithm (remember “e” = natural base) logex = ln x

Examples Evaluate: (Round to 3 decimals) 8. log 7 9. ln 0.25 = -1.386 = 0.845 9. ln 0.25 = -1.386 On TI-83: LOG button is base 10 and is to the left of 7 LN button is base e and is to the left of 4

Logarithm Inverse Properties

Examples Simplify the expressions.

Finding Inverses of Logarithms SAME Steps as Before!!! First, switch the x’s and y’s. Rewrite the logarithm equation as an exponential equation. Solve for y.

Examples x = log8y 8x = y y = 8x Find the inverse of the following functions. 14. y = log8 x x = log8y 8x = y y = 8x

Examples 15. y = ln (x – 10) x = ln(y – 10) ex = y – 10 y = ex + 10

Homework p.490 #16-64 evens

Logarithmic Functions (Day 2) (Graphing…yeah!) Algebra 2: Section 8.4 Logarithmic Functions (Day 2) (Graphing…yeah!)

Definition of Logarithm (Reminder)

Change of Base Formula Used to evaluate logs that are bases other than 10 or e. Or to punch logs of base other than 10 or e into the calculator (for graphing).

Graphs of Logarithmic Functions

Graphs of Logarithmic Functions y = logb(x – h) + k Asymptote: h (x = “h”) Domain: Range: If b>1, curve opens up If 0<b<1, curve opens down To graph: Show the asymptote Plot the x-intercept (calc or…..) find by setting y = 0 (will have to do for SEVERAL!!!) rewrite as an exponential equation Solve for x

How to write in Calculator

Examples State the asymptote, the domain, and the range of each function. 1. y = log1/2x + 4

Examples 2. y = log3(x – 2)

Homework p.491 #65-76 all State asymptote, x-intercept, domain, range Be sure asymptotes are graphed and labeled