7-3 Logarithmic Functions as Inverses

Objectives

To write and evaluate logarithmic expressions

To graph logarithmic functions

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Many even numbers can be written as power functions with base 2. In this lesson you will find ways to express all numbers as powers of a common base.

Essential Understanding The exponential function y = b y equals b  is one-to-one, so its inverse x = b x equals b  is a function. To express “y as a function of x” for the inverse, write y = log b x . y equals . log base b , x .

The exponent y in the expression b y b to the y  is the logarithm in the equation log b x = y . log base b , x equals y .  The base b in b y b to the y  and the base b in log b x log base b , x  are the same. In both, b ≠ 1 b not equal to 1  and b > 0 . b greater than 0 .

Since b ≠ 1 b not equal to 1  and b > 0 , b greater than 0 comma  it follows that b y > 0 . b to the y , greater than 0 .  Since b y = x b to the y , equals x  then x > 0 , x greater than 0 comma  so logb x is defined only for x > 0 . x greater than 0 .

Because y = b x y equals , b to the x  and y = y equals   log b x log base b , x  are inverse functions, their compositions map a number a to itself. In other words, b log b a b super log base b to the eh end super  for a > 0 eh greater than 0  and log b b a = a log base b . b to the eh , equals eh  for all a.

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