The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: 1000 = 10 × 10 × 10 = 103. More generally, if x = by, then y is the logarithm of x to base b, and is written y = logb(x), so log10(1000) = 3. Logarithms were introduced by John Napier in the early 17th century as a means to simplify calculations. They were rapidly adopted by navigators, scientists, engineers, and others to perform computations more easily, using slide rules and logarithm tables. Tedious multi-digit multiplication steps can be replaced by table look-ups and simpler addition because of the fact — important in its own right — that the logarithm of a product is the sum of the logarithms of the factors: The present-day notion of logarithms comes from Leonhard Euler, who connected them to the exponential function in the 18th century. The logarithm to base b = 10 is called the common logarithm and has many applications in science and engineering. The natural logarithm has the constant e (≈ 2.718) as its base; its use is widespread in pure mathematics, especially calculus. The binary logarithm uses base b = 2 and is prominent in computer science. Logarithmic scales reduce wide-ranging quantities to smaller scopes. For example, the decibel is a logarithmic unit quantifying sound pressure and voltage ratios. In chemistry, pH is a logarithmic measure for the acidity of an aqueous solution. Logarithms are commonplace in scientific formulae, and in measurements of the complexity of algorithms and of geometric objects called fractals. They describe musical intervals, appear in formulae counting prime numbers, inform some models in psychophysics, and can aid in forensic accounting. In the same way as the logarithm reverses exponentiation, the complex logarithm is the inverse function of the exponential function applied to complex numbers. The discrete...

...using only addition, subtraction and a table of squares. However it could not be used for division without an additional table of reciprocals. Large tables of quarter squares were used to simplify the accurate multiplication of large numbers from 1817 onwards until this was superseded by the use of computers.
Michael Stifel published Arithmetica integra in Nuremberg in 1544, which contains a table of integers and powers of 2 that has been considered an early version of a logarithmic table.
In the 16th and early 17th centuries an algorithm called prosthaphaeresis was used to approximate multiplication and division. This used the trigonometric identity
or similar to convert the multiplications to additions and table lookups. However logarithms are more straightforward and require less work. It can be shown using complex numbers that this is basically the same technique.
From Napier to Euler
John Napier (1550–1617), the inventor of logarithms.
The method of logarithms was publicly propounded by John Napier in 1614, in a book titled Mirifici Logarithmorum Canonis Descriptio (Description of the Wonderful Rule of Logarithms). Joost Bürgi independently invented logarithms but published six years after Napier. Johannes Kepler, who used logarithmtables extensively to compile...

...Maryann Crisci
Mrs. Cappiello
Algebra 2/Trig, Period 6
1 April 2012
Exponents and Logarithms
An exponent is the number representing the power a given number is raised to. Exponential functions are used to either express growth or decay. When a function is raised to a positive exponent, it will cause growth. However, when a function is raised to a negative exponent, it will cause decay. Logarithms work differently than exponents. Logarithms represent what power a base should be raised to in order to produce a specific given number. Logarithmic and exponential functions are often used together because they are inverse functions, and therefore “undo” one another. The natural exponential function and the natural logarithm are frequent examples. These functions have a base of the constant e, or Euler’s number. The natural exponential function is written as f(x)=ex. Its inverse, the natural logarithmic function, is written as f(x)=ln(x).
The decay of radioactive substances can be represented by using an exponential function. The mass of the radioactive substance will decrease as time passes, but the rate of decay and the mass of the substance will always remain directly proportional. The decay of radioactive substances can be expressed by the function m(t)=m0e-rt, where m(t) represents the mass remaining at time t, r represents the rate of decay, and m0 represents the initial mass. When the rate of decay is...

...up to a jumbo jet.
A fly's wing is maybe 0.1 inch long, and a jumbo jet wing might be 1000 inches long (about 80 ft). It would be pretty tough to put more than one insect on that graph - if you scale the wing length axis to fit on a sheet of paper, all the points for insect wings would be jammed up against one side.
Now, instead of plotting length, what if we plot the logarithm of length? There will be as much space on the graph between 0.1 inch and 1 inch as there is between 100 inches and 1000 inches, because
log(0.1) = -1
log(1) = 0
log(100) = 2
log(1000) = 3
So the graph will be much easier to read.
Logarithms are used in a lot of places to scale numbers when there's a big range between the smallest and the largest numbers of interest, which makes them easier to talk about.
y=yi x e^-kt
where:
y - different between temprature of body and the constant temp of room
yi - initial temprature difference of body and room
e - eulers number (2.718...)
t - time in mins
k - constant for that particular body (usually what u are trying to find out in class tasks)
using logarithms, newtons law can predict how how a body (such as cup of coffee) will be after any given period of time.
Example 1: A $1,000 deposit is made at a bank that pays 12% compounded annually. How much will you have in your account at the end of 10 years?
Explanation and Solution:
At the end of the first year,...

...CHAPTER 2
EXPONENTIAL AND LOGARITHMS FUNCTIONS (534)
2.1 Exponential functions and their graph
Definition
An exponential function with base b is defined by the equation
[pic] Or [pic] ([pic] [pic]and x is a real number)
The domain of any exponential function is the interval ([pic] The range is the interval [pic]
Example 1: Graph the exponential function [pic]
Example 2: Graph the exponential function [pic]
Properties of exponential functions
1. The domain of any exponential function is the interval ([pic]
2. The range is the interval [pic]
3. The [pic]intercept is (0,1).
4. The [pic]axis is an asymptote of the graph.
5. The graph of [pic]passes through the point [pic]
2.2 Logarithms functions and their graphs
Example 1: Graph the logarithms function [pic]
|x |f(x) |
|0.25 |-2 |
|0.5 |-1 |
|1 |0 |
|2 |1 |
|4 |2 |
Example 2: Graph the logarithms function [pic]
|x |f(x) |
|0.5 |2 |
|0.5 |1 |
|1 |0 |
|2 |-1 |
|4 |-2 |
2.3 Logarithms
An exponential function defined by y = b[pic], b > 0 and b...

...
6. Evaluate the logarithm without a calculator:
7. Solve the following equations:
8. Fill in the chart and graph:
x
1/4
1/2
0
2
4
8
16
9. A biologist is researching a newly-discovered species of bacteria. At time t = 0 hours, he puts one hundred bacteria into what he has determined to be a favorable growth medium. Six hours later, he measures 450 bacteria.
a. Assuming exponential growth, what is the growth constant "k" for the bacteria? (Round k to two decimal places.)
b. After 10 hours, how many bacteria will there be?
c. When will there be 10,000 bacteria?
10. A certain type of bacteria, given a favorable growth medium, doubles in population every 6.5 hours. Given that there were approximately 100 bacteria to start with, how many bacteria will there be in a day and a half?
Name:______________________________ Exponential and Logarithmic Functions.
1. Fill in the chart and graph:
x
-3
-2
-1
0
1
2
3
2. Graph Note:
3. Graph the following:
4.
5. Convert to log form
6. Evaluate the logarithm without a calculator:
7. Solve the...

...History of LogarithmsLogarithms were invented independently by John Napier, a Scotsman, and by Joost Burgi, a Swiss. Napier's logarithms were published in 1614; Burgi's logarithms were published in 1620. The objective of both men was to simplify mathematical calculations. This approach originally arose out of a desire to simplify multiplication and division to the level of addition and subtraction. Of course, in this era of the cheap hand calculator, this is not necessary anymore but it still serves as a useful way to introduce logarithms. Napier's approach was algebraic and Burgi's approach was geometric. The invention of the common system of logarithms is due to the combined effort of Napier and Henry Biggs in 1624. Natural logarithms first arose as more or less accidental variations of Napier's original logarithms. Their real significance was not recognized until later. The earliest natural logarithms occur in 1618. It can’t be said too often: a logarithm is nothing more than an exponent. The basic concept of logarithms can be expressed as a shortcut…….. Multiplication is a shortcut for Addition: 3 x 5 means 5 + 5 + 5 Exponents are a shortcut for Multiplication: 4^3 means 4 x 4 x 4 Logarithms are a shortcut for Exponents: 10^2 = 100. The present definition of the logarithm is the exponent...

...
PRINCE ALFRED COLLEGE
YEAR 10 ADVANCED MATHEMATICS
TEST 4: Part 2
Thursday 12-08-09
TOPIC: Indices (Exponents) & Logarithms & modelling
Name:
Pastoral Care Group: 10
Maximum mark
Your mark
Grade
% mark
Class
average %
60
Graphics calculators are permitted provided you answer the questions as asked
Show all working clearly.
Unless otherwise stated in the question, all numerical answers must be given exactly or to three significant figures.
You are advised to show all formulae used.
Please turn over
1. Simplify each of the following, expressing your answers without brackets or negative indices:
(a) . (b) (c)...

...Logarithm Base
IB Math SL
Type I Portfolio
Lisa Phommaseng
Logarithm Base
Consider the given sequences:
Log28, log48, log88, log 168, log328,…
Log381, log981, log 2781, log8181,…
Log525, log2525, log12525, log62525,…
:
:
:
,…
For the first sequence Log28, log48, log88, log 168, log328,… you are to find the next two terms of each of the sequences you would have to determine the pattern. As we can see, the value of the bases, 2,4,8,16,32 are increasing by squares of 2 (ex: 22= 4, 23=8, 24=16 etc.,). The two terms following log328 will be log648 and log1288. To find the expression for the nth term of each sequence, you would have to write the expression in the form of . First you would find log28 to begin to look for a pattern. A formula that could be useful to solve this would be logbx = . The variable b would be the base, x would be the number of the logarithm. Saying that log28 = x, we are able to determine that x is 3 because 23 equals 8. The same would follow for the next terms : log48= x, 8 is equivalent to 23 and the base of 4 is equivalent to 22 (Log28 =). So, to that term would be . You would continue on with the next and you would notice that the pattern is 3 over whatever the number that follows. Now we see that the nth term of this sequence is an=.
The next sequence Log381, log981, log 2781, log8181,…you would determine the following two terms as the one above. A we can see the terms are...