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Chapter 9: Problem 22

Simplify. $$ \log _{5} 125 $$

Short Answer

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Step by step solution

01

Recognize the Base and Argument

Identify the base of the logarithm, which is 5, and the argument, which is 125. We need to express 125 as a power of the base 5.
02

Express Argument as a Power of the Base

Rewrite the number 125 using the base 5. Since 125 is equal to 5 raised to the power of 3, we can write it as: \( 125 = 5^3 \)
03

Apply the Logarithm Property

Use the logarithm property, \( \log_b (b^x) = x \), which states that the logarithm of a base raised to an exponent is equal to the exponent. Thus, we have: \( \log_5 (5^3) = 3 \)
04

Simplify

The expression can be simplified to 3, because \( \log_5 (5^3) = 3 \).

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Key Concepts

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

Logarithmic Properties
Logarithmic properties are rules that simplify the process of evaluating logarithms. One important property is the power rule: when the argument of a logarithm is a power of the base, the logarithm can be simplified by removing the base and exponent. This property can be expressed as:
\ \( \log_b (b^x) = x \)
This means that if you have a logarithm with a base raised to a power, you can simplify the expression by just writing the exponent.
Exponents
Exponents are a shorthand way to represent repeated multiplication of a number by itself. In our example, we had to express 125 as a power of 5. This is because 125 can be written as 5 raised to the power of 3.
In exponent notation:
\ \( 5^3 = 125 \)
Breaking down such problems often involves identifying how many times the base multiplies to get to the argument.
Base and Argument in Logarithms
Understanding the base and the argument in logarithms is crucial for simplifying them. The base is the number you are repeatedly multiplying, and the argument is the result of those multiplications.
For instance, in \ \( \log_5 125 \), 5 is the base and 125 is the argument.
The goal often is to rewrite the argument as a power of the base to make simplification possible.
Simplification of Logarithmic Expressions
Simplification involves applying properties of logarithms to break down the expression to its simplest forms. In our example, we:
  • Identified the base (5) and argument (125)
  • Rewrote the argument as a power of the base (\ \( 125 = 5^3 \) )
  • Used the property \ \( \log_b (b^x) = x \) to simplify to the exponent
Finally, \ \( \log_5 125 \) simplifies to 3.
Simplification often boils down to repeatedly using the properties of logarithms until the expression cannot be reduced further.

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