# 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) , so:x = 35 Related:  MatemáticasMA304 Math Project

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 >>

Introduction to Logarithms In its simplest form, a logarithm answers the question: How many of one number do we multiply to get another number? Example: How many 2s do we multiply to get 8? Answer: 2 × 2 × 2 = 8, so we had to multiply 3 of the 2s to get 8 So the logarithm is 3 How to Write it We write "the number of 2s we need to multiply to get 8 is 3" as: log2(8) = 3 So these two things are the same: The number we multiply is called the "base", so we can say: "the logarithm of 8 with base 2 is 3" or "log base 2 of 8 is 3" or "the base-2 log of 8 is 3" Notice we are dealing with three numbers: the base: the number we are multiplying (a "2" in the example above) how often to use it in a multiplication (3 times, which is the logarithm) The number we want to get (an "8") More Examples Example: What is log5(625) ... ? We are asking "how many 5s need to be multiplied together to get 625?" 5 × 5 × 5 × 5 = 625, so we need 4 of the 5s Answer: log5(625) = 4 Example: What is log2(64) ... ? Answer: log2(64) = 6 Exponents In this way: log(100)

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

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

How to Safely Convert From One Unit to Another We can convert from km/h (kilometers per hour) to m/s (meters per second) like this: A kilometer has 1000 meters, and an hour has 3600 seconds, so a kilometer per hour is: 1000 / 3600 = 0.277... m/s How did I know to make it 1000 / 3600, and not 3600 / 1000 (the other way around)? The trick is to do the conversions as fractions! Example 1: Let's start with a simple example: convert 3 km to m (3 kilometers to meters). The system is: Write the conversion as a fraction Multiply Cancel any units that are both top and bottom You can write the conversion as a fraction that equals 1: 1000 m1 km = 1 And it is safe to multiply by 1 (does not affect the answer) so we can do this: 3 km x 1000 m1 km = 3000 km · m 1 km The answer looks strange! 3000 km · m1 km = 3000 m So, 3 km equals 3000 m. And the trick is to know that you will cancel when you finish, so make sure you write the conversion the correct way around (so you can cancel afterwards). Doing it wrong (with the conversion upside down) gets this: 1. 2.

Algebra - Exponential Functions Algebra (Notes) / Exponential and Logarithm Functions / Exponential Functions[Notes][Practice Problems][Assignment Problems] Algebra - Notes Let’s start off this section with the definition of an exponential function. Notice that the x is now in the exponent and the base is a fixed number. This is exactly the opposite from what we’ve seen to this point. To this point the base has been the variable, x in most cases, and the exponent was a fixed number. Before we get too far into this section we should address the restrictions on b. and these are constant functions and won’t have many of the same properties that general exponential functions have. Next, we avoid negative numbers so that we don’t get any complex values out of the function evaluation. the function would be, and as you can see there are some function evaluations that will give complex numbers. Now, let’s take a look at a couple of graphs. Note as well that we could have written in the following way, Properties of where . and . or .

Canada's Number 1 Math Website - Mathletics.ca - Love Learning Metric Numbers (See also Metric/Imperial Conversion Charts and Unit Converter) What is kilo, mega, giga, tera ... ? In the Metric System there are standard ways to talk about big and small numbers: "kilo" for a thousand, "mega" for a million, and more ... So we used kilo in front of the word meter to make "kilometer". And the abbreviation is "km" (k for kilo and m for meter, put together). Some more examples: Example: You put your bag on a set of scales and it shows 2000 grams, we can call that 2 kilograms, or simply 2 kg. Example: The doctor wants you to take 5 thousandths of a liter of medicine (a thousandth is one thousand times smaller), he is more likely to say "take 5 milliliters", or write it down as 5 mL. "kilo", "mega", "milli" etc are called "prefixes": Prefix: a word part that can be added to the beginning of another word to create a new word So, using the prefix "milli" in front of "liter" creates a new word "milliliter". Here we list the prefix for commonly used big and small numbers:

Basic Log Rules / Expanding Log Expressions Basic Log Rules / Expanding Logarithmic Expressions (page 1 of 5) Sections: Basic log rules, Expanding, Simplifying, Trick questions, Change-of-Base formula You have learned various rules for manipulating and simplifying expressions with exponents, such as the rule that says that x3 × x5 equals x8 because you can add the exponents. Log Rules: 1) logb(mn) = logb(m) + logb(n) 2) logb(m/n) = logb(m) – logb(n) 3) logb(mn) = n · logb(m) In less formal terms, the log rules might be expressed as: 1) Multiplication inside the log can be turned into addition outside the log, and vice versa. 2) Division inside the log can be turned into subtraction outside the log, and vice versa. 3) An exponent on everything inside a log can be moved out front as a multiplier, and vice versa. Warning: Just as when you're dealing with exponents, the above rules work only if the bases are the same. Expanding logarithms Log rules can be used to simplify expressions, to "expand" expressions, or to solve for values.

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

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