# Logarithmic and exponential functions - Topics in precalculus

Exponential functions Inverse relations Exponential and logarithmic equations Creating one logarithm from a sum THE LOGARITHMIC FUNCTION WITH BASE b is the function y = logb x. b is normally a number greater than 1 (although it need only be greater than 0 and not equal to 1). Note the following: • For any base, the x-intercept is 1. To see the answer, pass your mouse over the colored area. The logarithm of 1 is 0. y = logb1 = 0. • The graph passes through the point (b, 1). The logarithm of the base is 1. logbb = 1. Proper fractions. • The range of the function is all real numbers. • The negative y-axis is a vertical asymptote (Topic 18). Example 1. And here is the graph of y = ln (x − 2) -- which is its translation 2 units to the right. The x-intercept has moved from 1 to 3. Problem 1. This is a translation 3 units to the left. By definition: logby = x means bx = y. Corresponding to every logarithm function with base b, we see that there is an exponential function with base b: y = bx. Problem 2. and and Related:  Matemáticas

Logarithms: Introduction to "The Relationship" Purplemath offers a complete lessonon the topic you have selected.Try the lesson below! This lesson is not yet availablein MathHelp.com. Logarithms: Introduction to "The Relationship" (page 1 of 3) Sections: Introduction to logs, Simplifying log expressions, Common and natural logs Logarithms are the "opposite" of exponentials, just as subtraction is the opposite of addition and division is the opposite of multiplication. In practical terms, I have found it useful to think of logs in terms of The Relationship: On the left-hand side above is the exponential statement "y = bx". If you can remember this relationship (that whatever had been the argument of the log becomes the "equals" and whatever had been the "equals" becomes the exponent in the exponential, and vice versa), then you shouldn't have too much trouble with logarithms. (I coined the term "The Relationship" myself. Convert "63 = 216" to the equivalent logarithmic expression. Top | 1 | 2 | 3 | Return to Index Next >>

Working with Exponents and Logarithms What is an Exponent? What is a Logarithm? A Logarithm goes the other way. It asks the question "what exponent produced this?": And answers it like this: In that example: The Exponent takes 2 and 3 and gives 8 (2, used 3 times in a multiplication, makes 8) The Logarithm takes 2 and 8 and gives 3 (2 makes 8 when used 3 times in a multiplication) A Logarithm says how many of one number to multiply to get another number So a logarithm actually gives you the exponent as its answer: (Also see how Exponents, Roots and Logarithms are related.) Working Together Exponents and Logarithms work well together because they "undo" each other (so long as the base "a" is the same): They are "Inverse Functions" Doing one, then the other, gets you back to where you started: Doing ax then loga gives you x back again: Doing loga then ax gives you x back again: It is too bad they are written so differently ... it makes things look strange. going up, then down, returns you back again:down(up(x)) = x (and vice versa) And also:

Canada's Number 1 Math Website - Mathletics.ca - Love Learning Natural Logarithm -- from Wolfram MathWorld The natural logarithm is the logarithm having base e, where This function can be defined for This definition means that e is the unique number with the property that the area of the region bounded by the hyperbola , the x-axis, and the vertical lines and is 1. The notation is used in physics and engineering to denote the natural logarithm, while mathematicians commonly use the notation . denotes a natural logarithm, whereas denotes the common logarithm. There are a number of notational conventions in common use for indication of a power of a natural logarithm. (i.e., using a trigonometric function-like convention), it is also common to write Common and natural logarithms can be expressed in terms of each other as The natural logarithm is especially useful in calculus because its derivative is given by the simple equation whereas logarithms in other bases have the more complicated derivative The natural logarithm can be analytically continued to complex numbers as where is the complex modulus and

Logarithmic Functions The section covers: Half-Life problems can be found here. What is a Log and Why do we Need Them? I have to admit that logs are one of my favorite topics in math. I’m not sure exactly why, but you can do so many awesome things with them! We’ll soon see that Logs can be used to “get the variable in the exponent down” so we can solve for it. A slide rule was used (among other things) to multiply and divide large numbers by adding and subtracting their exponents. Definition of Logarithm Remember: A log is in exponent!!! Note that b is called the base of the log, and must be greater than 0 (so we don’t have to deal with complex numbers). The y in the log equation is called the argument and it must be greater than 0, again, to avoid complex numbers. To illustrate how these two equations are related, many times a “loop” is shown, which shows that b raised to the x equals y. or . Note: If there is no b next to the log, then the base is assumed to be 10. and , but we don’t need the 10. . as . 1. 2. 3.

Reflections of a graph - Topics in precalculus CONSIDER THE FIRST QUADRANT point (a, b), and let us reflect it about the y-axis. It is reflected to the second quadrant point (−a, b). If we reflect (a, b) about the x-axis, then it is reflected to the fourth quadrant point (a, −b). Finally, if we reflect (a, b) through the origin, then it is reflected to the third quadrant point (−a, −b). The distance from the origin to (a, b) is equal to the distance from the origin to (−a, −b). Example 1. Fig. 1 is the graph of the parabola f(x) = x2 − 2x − 3 = (x + 1)(x − 3). The roots −1, 3 are the x-intercepts. Fig. 2 is its reflection about the x-axis. Again, Fig. 1 is y = f(x). Fig. 3 is the reflection of Fig. 1 about the y-axis. The argument x of f(x) is replaced by −x. If y = f(x), then y = f(−x) is its reflection about the y-axis, y = −f(x) is its reflection about the x-axis. Problem 1. a) Sketch the graph of f(x). To see the answer, pass your mouse over the colored area. x2 + x − 2 = (x + 2)(x − 1). b) Write the function −f(x), and sketch its graph.

Parent Functions and Transformations - She Loves Math This section covers: Note that examples of transformations of Rationals can be found here, and more examples of transformations of Exponential and Log Functions can be found here. Also, Transformations of Trig Functions can be found here, and Transformations of the Inverse Trig Functions can be found here. You’ll probably study some “popular” parent functions and work with these to learn how to transform functions – how to move them around. The chart below provides some basic parent functions that you should be familiar with. I know we haven’t covered all these yet (sorry!) You’ll learn about the trigonometric parent functions and transformations here in the Graphs of Trig Functions and Transformations of Trig Functions sections, respectively. We haven’t really talked about end behavior, but we will in the Asymptotes and Graphing Rational Functions and Graphing Polynomials sections. Most of the time, our end behavior looks something like this: and we have to fill in the y part. instead of .

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