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1.5 Functions and Logarithms Golden Gate Bridge San Francisco, CA Photo by Vickie Kelly, 2004 Greg Kelly, Hanford High School, Richland, Washington
In other words, a function is one-to-one on domain D if: whenever A relation is a function if: for each x there is one and only one y. A relation is a one-to-one if also: for each y there is one and only one x.
To be one-to-one, a function must pass the horizontal line test as well as the vertical line test. one-to-one not one-to-one not a function (also not one-to-one)
Inverse functions: Given an x value, we can find a y value. Solve for x: Inverse functions are reflections about y = x. Switch x and y: (eff inverse of x)
is called the natural log function. is called the common log function. Consider This is a one-to-one function, therefore it has an inverse. The inverse is called a logarithm function. Two raised to what power is 16? Example: The most commonly used bases for logs are 10: and e:
In calculus we will use natural logs exclusively. We have to use natural logs: Common logs will not work. is called the natural log function. is called the common log function.
Properties of Logarithms Since logs and exponentiation are inverse functions, they “un-do” each other. Product rule: Quotient rule: Power rule: Change of base formula:
Example 6: $1000 is invested at 5.25 % interest compounded annually. How long will it take to reach $2500? We use logs when we have an unknown exponent. 17.9 years In real life you would have to wait 18 years. p*
Indonesian Oil Production (million barrels per year): Example 7: Use the natural logarithm regression equation to estimate oil production in 1982 and 2000. How do we know that a logarithmic equation is appropriate? In real life, we would need more points or past experience.