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Chapter 6: Problem 29

Find the exact value of each logarithm without using a calculator. $$ \log _{7} 49 $$

Short Answer

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Step by step solution

01

Understanding the Problem

Identify the base and the argument of the logarithm. In this case, the base is 7 and the argument (the number inside the logarithm) is 49.
02

Express the Argument as a Power of the Base

Express 49 as a power of 7. Since 49 is equal to 7 squared, we write it as: 49 = 7^2.
03

Apply the Logarithm Property

Use the property of logarithms that states: \(\log_b(b^k) = k\). In this case, \(\log_7(7^2)\).
04

Simplify the Logarithm

According to the property applied, \(\log_7(7^2) = 2\). So, the exact value is 2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithm Properties
Logarithm properties help us simplify and solve logarithmic equations. One crucial property is the Power Rule, which states \(\log_{b}(b^k) = k\). This property means if the argument (what's inside the logarithm) is a power of the base, the logarithm simplifies to that power. For example, in the exercise where \(\log_{7}(49)\), we found that 49 is \7^2\. Using the Power Rule, \(\log_{7}(7^2) = 2\). Always remember, these properties are key tools for breaking down logarithms into simpler components.
Logarithmic Functions
Logarithmic functions are the inverses of exponential functions. If you have an exponential function \(y = b^x\), its logarithmic counterpart is \(x = \log_{b}(y)\). This defines logarithms as the power to which a base must be raised to obtain a given number. For example, in the exercise, \(\log_{7}(49) = 2\) means 7 must be raised to the power of 2 to get 49. Understanding this relationship helps visualize what logarithms represent and aid in solving logarithmic equations effectively.
Base and Argument Identification
Identifying the base and argument in logarithms is fundamental. The base is the smaller subscript number, and the argument is the main number inside the logarithm. In the exercise \(\log_{7}(49)\), 7 is the base, and 49 is the argument. Recognizing these parts allows us to apply logarithmic properties correctly. The base indicates which number we repeatedly multiply, while the argument is the resultant number we aim to find the power for. Getting familiar with this identification step makes logarithmic problems simpler to work through from the start.
Exponentiation
Exponentiation involves raising a number (the base) to a power. This is key to solving logarithms as logarithmic functions are the inverses of exponentiation. For example, \(7^2 = 49\) shows how 7 raised to the power of 2 equals 49. By reversing this, \(\log_{7}(49) = 2\). We find the power needed to raise the base to get the argument. Understanding exponentiation helps explain why logarithms work as they do and solve equations effectively. Practice this relationship for better clarity on how logarithms and exponents interact.

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Most popular questions from this chapter

In Problems 79-84, graph each function using a graphing utility and the Change-of-Base Formula. \(y=\log _{4} x\)

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