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After reading this text and / or viewing the video tutorial on this topic you should be able to: explain what is meant by a logarithm. state and use the laws of logarithms. solve simple equations requiring the use of logarithms.
How do we decide what is the correct way to solve a logarithmic problem? The key is to look at the problem and decide if the problem contains only logarithms or if the problem has terms without logarithms.
In this booklet we will demonstrate how logarithmic functions can be used to linearise certain functions, discuss the calculus of the exponential and logarithmic functions and give some useful applications of them.
To solve equations which involve logarithms or exponentials we need to be aware of the basic laws which govern both of these mathematical concepts. We illustrate by considering some examples.
Simplify each of the following logarithmic expressions, giving the final answer as a single fraction. a) log 24 b) log 84 c) log 2 24 ( ) d) 5 1 log 125 1 2, 3 2, 3 4, 3 2 −
Every exponential function of the form f (x) bx, where b is a positive real number. =. other than 1, has an inverse function that you can denote by g(x) = logb x. This inverse function is called a logarithmic function with base b.
Example 2.3 Solve 15 = 8ln(3x) + 7. Solution: Subtract 7 from both sides and divide by 8 to get 11 4 = ln(3x) Note, ln is the natural logarithm, which is the logarithm to the base e: lny = log e y. Now, the equation above means 11 4 = log e (3x) so by the correspondence y = ax log a y = x, 3x = e11=4 which means x = 1 3 e11=4 3
