Problem 108 - Algebra and Trigonometry

Question

In Exercises 105–108, evaluate each expression without using a calculator. $$ \log (\ln e) $$

Step-by-Step Solution

Answer

The value of \(\log (\ln e)\) is 0.

Step 1: Evaluate the expression within the parentheses

Starting from inside the parentheses, evaluate \(\ln e\). Since by definition, the natural logarithm of \(e\) (i.e., \(\ln e\)) is 1, the given expression \(\log (\ln e)\), reduces to \(\log 1\).

Step 2: Evaluate the log expression

Now evaluate the logarithm of 1 (i.e., \(\log 1\)). Remembering that the base of the logarithm does not need to be specified when the base is 10, any number (including 10) raised to the power 0 gives 1. Therefore, \(\log 1\) is 0.

Key Concepts

Natural Logarithm Properties

When learning about logarithms, understanding the natural logarithm is crucial. The natural logarithm, denoted as \text{\(ln(x)\)}, is a logarithm with a base of Euler's number, \text{\(e\)}, approximately equal to 2.71828. Several properties of natural logarithms simplify solving algebraic problems. One essential property is that the natural logarithm of \text{\(e\)} itself is 1. This is because, by definition, a base raised to the power of 1 yields the base itself: \text{\(e^1 = e\)}.

An understanding of this property helps to solve the expression \text{\(\log (\ln e)\)}. Since \text{\(\ln e = 1\)}, it makes evaluating such expressions much simpler. Another key property to remember is that the natural logarithm of 1 is 0, \text{\(\ln 1 = 0\)}. This is because any number raised to the power of 0 equals 1, and therefore \text{\(e^0 = 1\)}. These properties are fundamental and can help students decipher more complex logarithmic equations.

Logarithmic Identities

Another piece of knowledge that's integral when working with logarithms is the set of logarithmic identities. These are rules that apply to all logarithms, no matter the base. For example, one such identity is \text{\(\log_b(1) = 0\)}, which tells us that the log of 1 to any base is 0, because \text{\(b^0 = 1\)} for any non-zero base \text{\(b\)}.

Another important identity is the change of base formula: \text{\(\log_b(x) = \frac{\log_k(x)}{\log_k(b)}\)}, which allows us to convert logs of different bases into a common base to simplify calculations. This identity wasn't directly used in our example but is a powerful tool when evaluating logarithms with bases other than 10 or \text{\(e\)}.

These identities form a backbone for solving log problems and can often simplify complicated expressions or allow one to change the expression into a more familiar base.

Base of Logarithms

Common and Natural Logarithms

When discussing the base of logarithms, there are two commonly used bases: 10, which is used for common logarithms \text{\(\log(x)\)}, and \text{\(e\)}, which is used for natural logarithms \text{\(\ln(x)\)}. It's essential to distinguish between these because the base determines how the logarithm behaves and how it should be evaluated.

For instance, a common logarithm does not require a base to be written explicitly because it's understood to be 10. On the other hand, natural logarithms always specify \text{\(e\)} as their base. Hence, knowing the proper base is key to evaluating expressions accurately.

In our exercise example, the absence of an explicit base in \text{\(\log(\ln e)\)} implies that the base is 10. This is why the final evaluation simplifies to 0, since any base raised to the power of 0 will yield 1, making \text{\(\log(1) = 0\)} regardless of the logarithm's base.Remember, recognizing the base can immediately tell you a lot about how to handle a logarithmic expression and lead to a quicker solution.