Problem 17 - Algebra and Trigonometry

Question

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

-6 \cdot \log N

Step 1: Applying the Power Rule of Logarithms

The power rule of logarithms states that the logarithm of a number raised to a power is equal to the power times the number. It is represented as \(\log_b(a^n) = n \cdot \log_b(a)\). Using the power rule of logarithms, the given expression \(\log {N^{-6}}\) can be written as \(-6 \cdot \log N\).

Step 2: Identify the relevant trigonometric identities

Based on the given expression or equation, identify which trigonometric identities (Pythagorean, double-angle, sum/difference, etc.) are applicable.

Step 3: Apply the identities and simplify

Apply the identified identities to transform the expression. Simplify step by step, combining like terms and reducing fractions where possible.

Step 4: Solve or evaluate

If solving an equation, isolate the trigonometric function and find the angle(s). If evaluating, compute the final numerical value.

Step 5: State the result

Express the final answer, including all solutions in the required domain if solving an equation.

Key Concepts

Power Rule of Logarithms

The power rule of logarithms is one of the foundational properties that makes working with logarithmic expressions more manageable. This rule applies when you have a logarithm of a number raised to a power. Using this rule, you can transform the expression into a product. This helps in simplifying or further manipulating the logarithmic expression.

For instance, if you take the expression \( \log_b(a^n) \), you can apply the power rule by moving the exponent, \( n \), in front of the logarithm. Thus, it becomes \( n \cdot \log_b(a) \). This not only simplifies the expression but also prepares it for other potential transformations as needed.

In the exercise provided, we have \( \log(N^{-6}) \). By applying the power rule, we bring the exponent \(-6\) in front of the log, thus rewriting the expression as \(-6 \cdot \log(N) \). This transforms a complex logarithmic expression into a simpler linear form that might be easier to handle or evaluate further.

Expand Logarithmic Expressions

Expanding logarithmic expressions involves using the properties of logarithms to break down more complex logs into simpler components. This is especially useful in algebra where simplification is key.

We use several properties of logarithms for expansion:

• **Power Rule:** As seen before, this helps to move exponents down in front of the log.
• **Product Rule:** Allows you to break down the log of a product into a sum of logs.
• **Quotient Rule:** Lets you express the log of a quotient as a difference of logs.

In our example, the expression was \( \log(N^{-6}) \). We've applied the power rule to expand this into \(-6 \cdot \log(N) \). Each term is simpler and more manageable and suits further algebraic manipulation or evaluation. Using these expansions, one can solve more complex logarithmic equations in steps by dealing with simpler parts.

Evaluate Logarithms Without Calculator

Evaluating logarithms by hand without a calculator is a useful skill, particularly when tackling basic exponential relationships. To efficiently evaluate logs without a tool, understanding common log values and properties becomes invaluable.

Here are some tips to help with manual calculations:

• **Knowledge of Base Values:** Being familiar with log values of numbers like 1, 10, and their multiples within common bases can streamline calculations.
• **Using Exponentials:** Recognize that logarithms are inversely related to exponentials, e.g., \( b^{ \log_b(a)} = a \).
• **Applying Logarithm Properties:** Utilizing rules like the power, product, and quotient rules allows you to break down complex logs and sometimes estimate them manually.

In scenarios where precise evaluation isn’t feasible by hand, estimating or simplifying the expression first allows for checking correctness or understanding the log's behavior. For the step-by-step solution, manual evaluation would still require knowing the specific value of \( \log(N) \), or being able to relate it to a function or constant that’s manageable without a calculator. This reinforces the importance of practice and familiarity with log properties and common logs.