Problem 33 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the expression without using a calculator.\(\log _{7} 7\)

Step-by-Step Solution

Answer

The evaluated expression \(\log _{7} 7\) is 1.

Step 1: Understand the call of the question

The key to solving this exercise is understanding that the expression \(\log_{7}7\) represents the power to which the base number (7) must be raised to produce the number given within the log (also 7).

Step 2: Apply Logarithmic Laws

When we apply the laws of logarithms it indicates that any value of 'a' where \(a > 0\), the result of \(\log_{a}a\) is equal to 1. Because in this case, you're trying to find the power to which the base (7) must be raised to result in the number 7 itself.

Step 3: Conclude your answer

Therefore, \(\log_{7}7\) equals 1. This is because 7 to the power of 1 gives you 7, satisfying the conditions of a logarithmic expression.

Key Concepts

Logarithmic Laws

Logarithmic laws are essential rules that simplify the manipulation and evaluation of logarithmic expressions. These laws are much like the laws of exponents, helping us understand and solve problems involving logarithms easily.
The most crucial law involved in this exercise is the identity law. The identity law states that for any base "a" where "a" is greater than zero, the logarithm of "a" to its own base is always 1. Mathematically, this is expressed as \(\log_{a}a = 1\).

• The reason behind this is that the logarithm asks the question: "To what power must we raise 'a' to obtain 'a'?" The answer is clearly 1, since any number raised to the first power is itself.

This law becomes a handy shortcut in evaluating logarithmic expressions without needing complex calculations. Understanding this fundamental relationship makes solving logarithmic problems more intuitive.

Logarithmic Expressions

A logarithmic expression is a mathematical expression involving a logarithm. It's a way of expressing how many times you need to multiply a base number to get another number. In notation, it's written as \(\log_{a}b\), where "a" is the base and "b" is the number you want to achieve through repeated multiplication of "a".

• The expression \(\log_{a}b\) essentially asks: "What exponent do we need for base 'a' to get 'b'?"
• So, if \(b = a^x\), then \(\log_{a}b = x\).

It's important to become familiar with how to read and understand these expressions.
This helps in decoding the mathematical implications of what the logarithm is asking. The expression \(\log_{7}7\) illustrates this clearly, as it asks how many times we must multiply 7 by itself to get 7, which, unsurprisingly, is once.

Exponents

Exponents are a way to express repeated multiplication of a number by itself. If we have a number "a" raised to the power of "x", it means we're multiplying "a" by itself "x" times. This concept is fundamental in understanding both exponents and logarithms.

• For instance, \(a^1 = a\), illustrating that any number raised to the power of 1 is the number itself.
• This principle is pivotal to understanding the identity law of logarithms, such as \(\log_{a}a\).

The close relationship between exponents and logarithms lies in their inverse nature. While exponents represent multiplication, logarithms are about finding the necessary power from a given multiplication.
This interconnectedness is key to grasp, making it easier to transition between and solve problems involving both exponents and logarithms.