Problem 63 - Precalculus : Building Concepts and Connections

Question

In Exercises \(53-64,\) simplify each expression. Assume that each variable expression is defined for appropriate values of \(x .\) Do not use a calculator. $$\log _{a} \sqrt[5]{a^{2}}, a > 0, a \neq 1$$

Step-by-Step Solution

Answer

The simplified form of the expression is \( \frac{2}{5} \)

Step 1: Change base

Change the base of the logarithm using the power rule of logarithms, which says that \(\log_b a^n = n \log_b a\). This gives us: \( \frac{1}{5} \log_a (a^2) \)

Step 2: Apply the property of logarithms.

The log base \(a\) of \(a\) equals 1, giving us \( \frac{1}{5} \cdot 2 = \frac{2}{5} \)

Key Concepts

Logarithm Properties

Logarithms are like detectives of the math world; they help us figure out what mystery number was raised to a certain power to achieve another number. But to do that, they follow certain rules or properties that make simplifying them easier. Some of these properties include:

• The Product Rule: which tells us that the log of a product is equal to the sum of the logs of its factors, making multiplication problems a lot less scary.
• The Quotient Rule: which is like the product rule's twin but for division, showing us the log of a quotient is the difference between the logs of the numerator and denominator.
• The Power Rule: is the superhero of logs, allowing us to take the exponent and make it a multiplier, which is a huge simplification.
• Plus, there's the rule that a log with the same base and argument, like \(\text{log}_a(a)\), is always 1 because any number to the power of 1 is itself.

When you're tackling a logarithm without a calculator, these properties are your best buddies. They let you break down complex expressions into bite-sized pieces that are much easier to chew!

Power Rule of Logarithms

The power rule of logarithms is like having a secret code that can simplify any intimidating-looking logarithmic expression. When you see something like \(\text{log}_b(a^n)\), you can use this rule to rewrite it as \(n \times \text{log}_b(a)\). That means the nasty-looking exponent 'n' comes down to play on ground level and multiplies the log.

This rule makes it simple to tackle logs with large exponents or expressions within a root, just as we did with the logarithm \(\text{log}_a \text{ of } \text{the fifth root of } a^2\). By applying the power rule, we brought down the exponent, turning it into \( \frac{1}{5} \text{log}_a (a^2)\). And voilà, what was once a complex expression is now something much simpler that we can work with without the need for a calculator!

Logarithmic Expressions Without Calculator

When it comes to solving logarithmic expressions without a calculator, it's like being a math explorer without a map - you need to rely on your wits and the tools you've been taught. The properties of logarithms are your compass. Knowing them allows you to navigate through tricky problems with confidence.

Let's take the expression \( \text{log}_{a} \text{ of } \text{the fifth root of } a^2 \) from our exercise as an example. Remembering that any log where the base and the argument are the same number is just a fancy way of writing '1,' we don't need to compute anything crazy. After applying our power rule, we end up with \(\frac{1}{5} \times 2\), which simplifies to \(\frac{2}{5}\). That's it! By understanding the rules and properties of logs, you don't need a calculator to crack the code of logarithmic expressions.