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Chapter 9: Problem 14

Simplify. $$ \log _{10} 100 $$

Short Answer

Expert verified
\( \text{log}_{10}(100) = 2 \)

Step by step solution

01

Understand the Logarithm Definition

Recall that the logarithm \(\text{log}_{b}(a)\) answers the question: 'To what power must \(b\) be raised, to obtain \(a\)?'
02

Set Up the Logarithmic Expression

In this problem, \(b = 10\) and \(a = 100\). Therefore, we need to find \(\text{log}_{10}(100)\).
03

Convert the Logarithm to an Exponential Form

We are looking for \(x\) such that \(10^x = 100\).
04

Solve the Exponential Equation

\text{Since } 10^2 = 100, \[ x = 2 \].
05

State the Simplified Logarithmic Value

Thus, \( \text{log}_{10}(100) = 2 \).

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Key Concepts

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

Logarithmic Definition
Understanding the definition of a logarithm is key to solving problems related to it. A logarithm, written as \(\text{log}_b(a)\), essentially asks the question: 鈥楾o what power must \b\ be raised to get \a\?鈥�.
For example, if you see \( \text{log}_{10}(100)\), it asks: 鈥榃hat power should we raise 10 to, in order to achieve 100?鈥�
Exponential Form
To simplify logarithmic expressions like \( \text{log}_{10}(100)\), we can convert them into their exponential form. The exponential form of a logarithmic equation \( \text{log}_b(a) = c \) is \( b^c = a \). This means:
  • For \( \text{log}_{10}(100)\), we write \( 10^x = 100 \).
It helps to recognize that the logarithm and exponential forms are two sides of the same coin. They powerfully connect multiplication and exponentiation.
Solving Exponential Equations
Solving exponential equations requires understanding the relationship of the base and exponent. For the equation \( 10^x = 100 \), we know from basic exponent rules that:
  • \( 10^2 = 100 \).
Thus, the value of \x\ is 2 in this context. Therefore, \( \text{log}_{10}(100) = 2 \). Always verify your solution by back-substituting to the original logarithmic form to ensure correctness.

This approach simplifies the process of solving logarithmic expressions by breaking them down into more understandable and solvable pieces.

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