Problem 17 - College Algebra

Question

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log N^{-6} $$

Step-by-Step Solution

Answer

The expanded form of the logarithmic expression \(\log N^{-6}\) is -6 \(\log(N)\)

Step 1: Identify the given expression

The given logarithmic expression is \(\log N^{-6}\). Here, N is the base of the logarithm and -6 is the exponent.

Step 2: Apply the property of logarithm

To simplify the expression, we use the property \(\log(a^n) = n \log(a)\). By applying this rule to the given expression, we continue with -6 \(\log(N)\).

Step 3: Final Result

By applying the rule, we arrived at the final expanded form of the given expression which is -6 \(\log(N)\)

Key Concepts

Logarithmic Expressions

When we talk about logarithmic expressions, we are referring to mathematical expressions involving logarithms. A log function, represented as \( \log_b (x) \), describes how many of a certain number (the base \( b \)) must be multiplied together to get another number \( x \). Understanding how to interpret and manipulate these expressions is essential to mastering concepts in algebra and calculus.

Take for instance the logarithmic expression \( \log N^{-6} \) from our exercise. It's compact but can be broken down using properties of logarithms to make its meaning clearer. Knowing that a logarithm essentially asks the question 'to what power must we raise the base to get the number in question', helps us to intuitively understand the components of this expression — the base, which is understood to be 10 if not otherwise specified, and the number, \( N^{-6} \) in this case, which is raised to a power.

Expanding Logarithms

Expanding logarithms is a process that involves rewriting a log of an exponentiated term into a product of simpler terms. This is made possible by the properties of logarithms that interconnect multiplication, division, and exponentiation.

The key property used for expansion in our example is the Power Rule of logarithms, which states that \( \log(a^n) = n \cdot \log(a) \). To see this in action, we applied this rule to our expression \( \log N^{-6} \) to turn it into \( -6 \cdot \log(N) \), effectively pulling the exponent out in front of the log as a coefficient. This expanded form is not only more comprehensible but also primed for further simplification or evaluation. Notably, in more complex scenarios, properties addressing products and quotients within logarithms can also be applied to expand expressions further.

Evaluating Logarithms

Evaluating logarithms means finding the value of a logarithmic expression as accurately as possible, which sometimes can be done without a calculator. After expanding a log expression, we may be able to simplify it further or even find its exact value if we are dealing with known logarithms, such as \( \log(10) \) or \( \log(1) \) which are respectively 1 and 0.

In our exercise example, however, we encountered an expression \( -6 \cdot \log(N) \) after expanding. Without knowing the value of \( N \), we cannot fully evaluate this log expression to a single number. But remember, understanding the transformative steps of evaluation is powerful. It prepares you for instances when you will need to insert specific numbers or variables into your expanded logs to find concrete solutions to real-world problems.