The Natural logarithm and e

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The Natural logarithm and e IB SL/HL Adrian Sparrow www.ibmaths.com

Learning outcome: you should be able to solve equations by using the relationship between the natural logarithm and e.

Natural logarithms A logarithm to the base e is called a natural logarithm. loge x is written as ln x Point out that the domain of y = ln x is the set of positive real numbers. In other words, ln x is not defined for negative values of x (or x = 0). This follows from the fact that ex is always positive. The range of ln x, on the other hand, is the set of all real numbers. The graph of y = ln x is asymptotic to the y-axis. So, ln x is the inverse function of ex

Using the natural log - ln Without using a calculator find the value of: Use a calculator to find: = 3 =0 = 4 =1 = =2 = -3 = = -1 = n

The laws of natural logarithms It is also helpful to realise that ln ex = x

Finding a missing index using logarithms Find x to 2 decimal places using trial and error. This process can be used with any base log, even the natural log. Far too complicated ... Take a log of both sides Now try these, answers to 2 d.p. Use the power rule

Remember the base of a natural log is e. Find x, if Find x, if Remember the base of a natural log is e. Take a natural log of both sides. Rearrange in index form. Use the power rule. Find x in each of the following: Find x in each of the following: