Problem 29 - Intermediate Algebra

Question

In this Study Set, assume that all variables represent positive numbers and \(b \neq 1\). Evaluate each expression. See Example \(1 .\) $$ \log _{3} 3^{7} $$

Step-by-Step Solution

Answer

The expression evaluates to 7.

Step 1: Understanding the Property of Logarithms

We start by recognizing the property of logarithms that states \( \log_b b^x = x \). This property reveals that the logarithm with base \(b\) of \(b\) to the power of some number \(x\) reduces to \(x\).

Step 2: Apply the Property to the Expression

In our given expression, \( \log_{3} 3^{7} \), the base \(b\) is 3 and the exponent \(x\) is 7. We apply the logarithmic property: \( \log_3 3^7 = 7 \).

Step 3: State the Result

Thus, based on the logarithmic property, the value of the expression \( \log_{3} 3^{7} \) evaluates to 7.

Key Concepts

Properties of Logarithms

Logarithms have some interesting rules that can simplify complex expressions and make it easy to solve problems. One key property is the power rule for logarithms, which helps in compressing exponential terms. The power rule states that the logarithm of a number with an exponent is simply the exponent itself: \( \log_b b^x = x \).
This is because the logarithm answers the question: "To what exponent must the base \( b \) be raised, to yield this number?" When you look at the expression \( \log_3 3^7 \), it follows this rule straightforwardly since the base and the base of the exponent are the same (both are 3). The 3 is raised to the 7th power, making the result simply 7.
This property is very useful because it directly cancels out the base part of the operation, simplifying things enormously. In test problems, you'll often find these situations, so remembering this property can save you time and effort.

Exponents

Exponents are a shorthand way to describe repeated multiplication of a number. For instance, when you see \(3^7\), it means you multiply 3 by itself 7 times: \(3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3\).
This notation helps us work with very large or small numbers easily. Since reading and writing out whole sequences of multiplication can be cumbersome, exponents streamline this process.

• If you understand how exponents work, you can then understand that an expression like \( \log_3 3^7 \) is asking what power we must raise 3 to, to get \(3^7\) again.
• In general, exponents are used in many areas of mathematics, from the simple arithmetic you do daily to complex calculus operations.

Understanding exponents is crucial to not just solving problems like the ones involving logarithms, but also for broader mathematical literacy.

Logarithmic Expressions

Logarithmic expressions might seem complex at first, but they are very practical once you get a handle on them. These expressions involve logarithms, which are essentially the inverse operations of exponentiation.
For example, if you have \( b^x = y \), taking the logarithm of both sides, you get \( \log_b y = x \). This means that the logarithm tells you what power you need to raise the base \( b \) to in order to get the number \( y \).

• The expression \( \log_3 3^7 \) is a great demonstration because it directly correlates the concept of powers with that of logarithms: finding the power to which 3 must be raised to return to itself after being exponentiated.
• This particular expression is simplified by using the properties of logarithms, which allow it to resolve straightforwardly to the exponent itself.

Mastering these expressions helps in solving equations that involve exponential growth or decay, understanding sound intensity in decibels, and even calculating pH in chemistry. Once you're comfortable with basic logs, you can solve increasingly challenging problems with ease.