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Chapter 12: Problem 10

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

Short Answer

Expert verified
The simplified form of \(\text{log}_{10}{100}\) is \(2\).

Step by step solution

01

- Understand the Logarithmic Expression

Identify the given logarithmic expression. In this case, it is \(\text{log}_{10}{100} \). Your goal is to simplify this expression.
02

- Rewrite the Logarithm as an Exponent

Recall the definition of a logarithm: \(\text{log}_{b}{a} = c \) if and only if \({b}^{c} = a\). Applying this definition, we can write: \(\text{log}_{10}{100} = c \) implies \({10}^{c} = 100\).
03

- Find the Exponent

Determine which power of 10 equals 100. Note that \({10}^{2} = 100\). Therefore, \({10}^{c} = 100\) implies \(c = 2\).
04

- Simplify the Expression

Based on the previous step, replace \(c\) with \(2\). 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.

Logarithm Definition
A logarithm is a mathematical function that helps us determine the power to which a number (called the base) must be raised to obtain another number.
In simpler terms, if we have \(\text{log}_{b}{a} = c\), then \({b}^{c} = a\).
Think of it like this: 'logarithm' is the question and 'exponent' is the answer. So, in our example, when we see \(\text{log}_{10}{100}\), we're asking '10 raised to what power gives us 100?'
Knowing this definition is crucial to solving any logarithmic expression, as it turns the problem into a more familiar form, like an exponential equation.
Base 10
In many logarithmic problems, the base is 10. This is so common that 'log base 10' has a special notation: 'log' without a subscript.
For instance, \(\text{log}_{10}{100}\) can simply be written as \( \text{log} 100\).
Base 10 logarithms are also known as 'common logarithms,' and they are widely used in science and engineering.
They simplify calculations and make it easier to work with large numbers. For instance, the logarithm of 10,000 is simply 4 because \( \text{log}_{10} {10,000} = 4\) since \( {10}^{4} = 10,000 \).
Exponential Form
To solve logarithmic expressions, convert them to an exponential form.
For example, \(\text{log}_{10}{100} \) becomes \({10}^{c} = 100 \). This states that 10 must be raised to the power of 'c' to result in 100.
Next, determine 'c'. Since \( {10}^{2} = 100 \), it follows that \(\text{log}_{10} {100} = 2\).
This step-by-step transformation from logarithmic to exponential form and back to a simplified number is a key technique for any logarithmic simplification task.
  • Start by identifying the logarithmic expression.
  • Rewrite it according to the logarithm definition.
  • Solve for the exponent in the exponential equation.

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Most popular questions from this chapter

To prepare for Section \(12.5,\) review graphing functions and finding domains of functions. Graph. \(f(x)=\sqrt{x}-3\)

For each function given below, (a) determine the domain and the range, (b) set an appropriate window, and (c) draw the graph. $$ f(x)=3.4 \ln x-0.25 e^{x} $$

For each function given below, (a) determine the domain and the range, (b) set an appropriate window, and (c) draw the graph. $$ f(x)=2 x^{3} \ln x $$

Express as an equivalent expression that is a single logarithm and, if possible, simplify. $$\log _{a}(2 x+10)-\log _{a}\left(x^{2}-25\right)$$

Use the properties of logarithms to find each of the following. $$\log _{3}(9 \cdot 81)$$
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