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Chapter 9: Problem 24

Find the exact value. See Examples 2 and \(6 .\). \(\log 10^{7}\)

Short Answer

Expert verified
The exact value of \(\log 10^{7}\) is 7.

Step by step solution

01

Understand the Problem

We need to find the value of the logarithmic expression \(\log 10^{7}\). This involves using properties of logarithms to simplify the expression.
02

Recall the Logarithm Property

Recall the property of logarithms: \(\log b^{x} = x \log b\). In this problem, \(b=10\) and \(x=7\).
03

Apply the Logarithm Property

Using the property from Step 2, substitute into the expression: \(\log 10^{7} = 7 \log 10\).
04

Evaluate \(\log 10\)

Remember that \(\log 10\) is the logarithm of 10 with base 10, which equals 1. Therefore, \(\log 10 = 1\).
05

Calculate the Final Answer

Substitute the value of \(\log 10\) from Step 4 into the equation from Step 3: \(\log 10^{7} = 7 \times 1 = 7\).

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Key Concepts

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

Properties of Logarithms
Logarithms come with a set of properties that make them incredibly useful in mathematics. One critical property, used in the exercise, is the power rule:
  • For any positive number \( b \) and real number \( x \), the property states: \( \log_b (b^x) = x \).
  • This property simplifies expressions like \( \log 10^7 \) to 7.
  • It works because when you take a logarithm, you are essentially finding an exponent.
This power rule tells us how the logarithm of a number with an exponent can be simplified to focus on the exponent itself. By using this property, we can efficiently solve and simplify many logarithmic expressions, especially those involving powers.
Logarithmic Expressions
Logarithmic expressions involve operations with logarithms. These expressions can often be simplified using the properties of logarithms.
  • An expression like \( \log 10^7 \) can seem complex, but with certain properties, it becomes simpler.
  • The key is to break down the expression using properties such as the power rule and base calculations.
When working with such expressions, it's essential to apply these properties correctly. The expressions can often be re-arranged and calculated with ease, making it important to understand the properties deeply.
Base 10 Logarithms
Base 10 logarithms, commonly referred to as common logarithms, are extremely useful. They have a specific base of 10, which simplifies many calculations.
  • For example, \( \log 10 \) equals 1, since 10 is raised to the power of 1 to get 10.
  • Because of this easy calculation, base 10 logarithms are often used in scientific and engineering contexts.
  • The property \( \log 10^x = x \) is particularly easy to apply since we know the log of the base equals 1.
Understanding base 10 logarithms helps in simplifying and computing logarithmic expressions efficiently. They are a key tool for translating exponential information into digestible terms.

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Most popular questions from this chapter

Solve. See Example 4. $$ \log _{3} \frac{1}{27}=x $$

Write each as an exponential equation. See Example 1. $$ \log _{11} \sqrt[4]{11}=\frac{1}{4} $$

Solve each system of equations. See Section 4.1. $$ \left\\{\begin{array}{c} {5 x+y=5} \\ {-3 x-2 y=-10} \end{array}\right. $$

Graph each function by finding ordered pair solutions, plotting the solutions, and then drawing a smooth curve through the plotted points. $$ f(x)=\log x $$

Solve each system of equations. See Section 4.1. $$ \left\\{\begin{array}{r} {x+2 y=-4} \\ {3 x-y=9} \end{array}\right. $$
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