Exercise 25 Page 576

We are given the following approximations. ln 2 &≈ 0.69315 ln 3 &≈ 1.0986 Recall that ln x represents the natural logarithm of x. The natural logarithm is a logarithm with base e, which is a irrational constant called Euler's number. ln x= y ⇔ x=e^y The value of e is approximately 2.72. This means that e is greater than 2 and less than 3. 2< e < 3 Now, if we calculate a logarithm of a number that is less than the base of the logarithm, then the value of the logarithm must be less than 1. This is because we need to raise the base to a power of less than 1 to get a smaller number. e> 2 ⇒ ln 2 < 1 On the other hand, if we calculate a logarithm of a number that is greater than the base of the logarithm, then the value of the logarithm must be greater than 1. Remember, we need to raise the base to a power greater than 1 to get a bigger number. e < 3 ⇒ ln 3 > 1 Finally, let's use the definition of a natural logarithm to find for what value of x the following equation is true. ln x= 1 ⇔ x=e^1 We can see that the value of x is e. For all logarithms, the logarithm is equal to 1 only if the base and the number of which we evaluate a logarithm are the same.