7-1: Exponential Review and Logarithms Basics Unit 7: Exponents/Logarithms English Casbarro.

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7-1: Exponential Review and Logarithms Basics Unit 7: Exponents/Logarithms English Casbarro

Recall from Math 2: Growth adds to 100% and then change to a decimal for the growth factor. Decay subtracts from 100% and then change to a decimal for the decay factor. Ex. 3.4% growth  = 103.4% the growth factor is: Ex. 1.7% decay  100 – 1.7 = 98.3% the decay factor is: 0.983

Exponential Growth and decay Exponential Growth and Decay can be modeled by the following : A(t) = a(b) t The final amount The initial amount Growth/Decay Factor  ± 100, then Change to a decimal Number of time periods

Example 2: Between 2010 and 2013, the population of a city decreased at a rate of 3.9% per year. Its population in 1990 was 872,613. Fill in the chart for the Years listed below. Year Population

Recall: Finding Inverses If f and g are functions such that f(g(x)) = x and g(f(x)) = x, then f and g are inverses of each other. The notation is given as such: If f is the original function, then f -1 is the inverse of f. If f and g are inverses of each other, then the domain of f is the same as the range of g and vice versa.

Basic Inverses  Ex.  Domain: Range:  The inverse of is: Notice how the points have switched the x’s and y’s.  The domain is.  The range is.

Example 1: Find the inverse,  Write out the functions using a y instead of function notation.  Switch the x and the y.  Solve for y. (divide everything by 3) – 2 or

Example 2: Find the inverse.

Example 3: Find the inverse: f(x) = 1.5 x + 4 Y = 1.5 x + 4 x = 1.5 y + 4 (x – 4) = 1.5 y y = log 1.5 (x – 4) We will re-visit this problem for graphing once we finish the properties of logarithms. Rewrite this in Log form to get y by itself

Basic Logarithm Rules  Logs can be in any base  The “log” button on your calculator is base 10. It is called a “common log” (notice that above it shows 10 x)  The “ln” button on your calculator is base e. (recall: e = … also above the button: e x )

Example 4: a.Write 5 3 = 125 in logarithmic form b.Write log 3 81 = 4 in exponential form Example 5: Exponential Form 2 5 = 32 Logarithmic Form Complete the table.

Example 6: Solve 10 x = 85 1.Re-write as a log. 2. Use a calculator.