The Logarithm as Inverse Exponential Function Recall: If y is a one to one function of x, to find the inverse function reverse the x’s and y’s and solve.

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Presentation transcript:

The Logarithm as Inverse Exponential Function Recall: If y is a one to one function of x, to find the inverse function reverse the x’s and y’s and solve for y in terms of x. Recall or but … If x is a positive number then the logarithm of x to the base b denoted is the number y such that that is

Converting between logs and exponents Recall: Using this definition – evaluate the following Plot the graph of for Do the same for

Graphs of

If x is a positive number then the logarithm of x to the base b denoted is the number y such that that is Properties of the logarithmic function 1.It is defined and continuous for all x > 0 (restricted domain) 2.The y-axis is a vertical asymptote (no horizontal asymptotes) 3.The x-intercept is (1, 0); there is no y-intercept 4.If then and if then and 5If then is increasing if then is decreasing Note!

Logarithmic Rules (Know!) Identities: Equality Rule Product Rule Power Rule Quotient Rule Change of Base Rule Remember! Logarithms Are Exponents

Proofs for Other Rules Power Rule Change of Base Rule ← Definition of Logarithm

Solving Exponential Equations 1. Solve: 2.Given 50 grams of a radioactive substance with a half life of 15 days, how long until only 20 grams remain? 3.Given an initial population of 7000 and a growth rate of 6% per year, when will the population reach 10,000? Common Logs versus Natural Logs