Problem 39 - Algebra and Trigonometry

Question

In Exercises 21–42, evaluate each expression without using a calculator. $$ \log _{5} 5^{7} $$

Step-by-Step Solution

Answer

The value of \( \log _{5} 5^{7} \) is 7.

Step 1: Understanding the Expression

In this expression \( \log _{5} 5^{7} \), '5' is the base of the logarithm and '5^7' is the argument. In other words, this expression is asking 'To what power must we raise 5 in order to obtain \( 5^7 \)?' The answer can be deduced by understanding the basic property of logarithms.

Step 2: Applying Logarithmic Properties

According to a key property of logarithms, the log of a quantity (say 'n') to the same base 'n' is always equal to 1. This means that \( \log _{n} n = 1 \). Furthermore, when the argument is 'n' raised to an exponent (say 'k'), this exponent can be brought to the front in the form of multiplication due to another property of logarithms: \( \log _{n} n^k = k \times \log _{n} n \). Applying these properties to our original expression results in \( 7 \times \log _{5} 5 \).

Step 3: Solving the Expression

Now we can easily solve the expression. The log of 5 to the base 5 equals 1 (i.e., \( \log _{5} 5 = 1 \)), which simplifies our expression to \( 7 \times 1 \). So, \( \log _{5} 5^{7} = 7 \).

Key Concepts

Exponentiation

Exponentiation is a fundamental mathematical operation where a number, known as the base, is multiplied by itself a certain number of times. The number of times the base is used as a factor is called the exponent. For instance, in the expression \( 5^7 \), the number 5 is the base, and 7 is the exponent. This expression represents \( 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \). Each repetition of multiplication represents the increasing power of the base.

Exponents can be a powerful tool in simplifying large and complex calculations. They allow us to express large numbers compactly and can also be used to demonstrate growth, such as in exponential growth models in biology or finance. Understanding and manipulating exponents is crucial when working with logarithms, as you'll often encounter expressions where the base and the exponent need to be considered together.

Logarithms

Logarithms are the inverse operation of exponentiation. When you have an exponential equation like \( b^y = x \), the logarithm answers the question: "To what power must the base \( b \) be raised to get \( x \)?" If we denote this relationship as \( y = \log_b{x} \), we can use logarithms to solve for \( y \).

Logarithmic functions have several properties that make them highly useful in mathematics:

• \( \log_b{b} = 1 \): because any number raised to the power of 1 is itself.
• \( \log_b{1} = 0 \): as any non-zero number raised to the power of 0 is 1.
• \( \log_b{b^k} = k \log_b{b} \): a crucial property used in the exercise provided, which shows how exponents can be extracted from logarithmic expressions.

By understanding these properties, one can transform complex mathematical problems into simpler operations. They have powerful applications in science and engineering, like in measuring sound intensity (decibels) and calculating complex interest.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operations that represent a value or relationship. These expressions can be as simple as \( 2 + 3 \), or as complex as \( \log_5{5^7} \). When working with expressions, it is important to understand the order of operations, commonly remembered by the acronym PEMDAS:

• Parentheses
• Exponents (i.e., powers and roots)
• Multiplication and Division
• Addition and Subtraction

The ability to interpret and simplify mathematical expressions is a fundamental skill necessary across all fields of mathematics.

Moreover, knowing how to manipulate these expressions, such as by using the properties of logarithms and exponents, will allow you to solve problems more efficiently. This is highlighted in our example exercise, where breaking down the expression \( \log_5{5^7} \) requires understanding the interaction between logarithmic properties, exponentiation, and basic arithmetic operations. Understanding these principles allows for seamless conversion from complex expressions to simpler numerical results.