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Chapter 9: Problem 6

Write each as an exponential equation. See Example 1. $$ \log _{10} 10=1 $$

Short Answer

Expert verified
The exponential equation is \(10^1 = 10\).

Step by step solution

01

Understanding the Problem

We are given the logarithmic equation \( \log_{10} 10 = 1 \). Our task is to write this equation in an exponential form.
02

Identifying the Base

In the given equation \( \log_{10} 10 = 1 \), the base of the logarithm is \( 10 \). The base remains the same in the exponential form.
03

Interpreting the Logarithmic Equation

The equation \( \log_{10} 10 = 1 \) means 'What power should 10 be raised to in order to get 10?'. The answer to this is 1.
04

Constructing the Exponential Equation

Based on the logarithmic interpretation, we can translate \( \log_{10} 10 = 1 \) to the exponential form \( 10^1 = 10 \).

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Key Concepts

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

Logarithmic Equations
Logarithmic equations involve logarithms, which are the opposite of exponentiation. When you see a logarithmic equation, such as \( \log_{b} a = c \), it literally means "what power must the base \( b \) be raised to, in order to yield the number \( a \)?" This type of equation helps in determining unknown exponents.
  • The base \( b \), the result \( a \), and the exponent \( c \) all play crucial roles.
  • Logarithmic equations can be rewritten in exponential form, which is handy for solving them.
  • For example, \( \log_{10} 10 = 1 \) implies that 10 raised to the power of 1 equals 10.
By converting logarithmic equations to exponential form, students can more easily handle and solve complex equations. This technique is essential for mastering algebra and understanding how different mathematical operations relate to one another.
Exponentiation
Exponentiation is a mathematical operation that involves raising a number, known as the base, to the power of an exponent. In the expression \( b^c = a \), \( b \) is the base, \( c \) is the exponent, and \( a \) is the result.
  • Exponentiation is fundamental in many areas of mathematics, from basic arithmetic to advanced calculus.
  • It is the backbone of working with powers, especially when solving logarithmic and exponential equations.
  • For example, \( 10^1 = 10 \) illustrates a simple case of exponentiation, showing how a number is raised to the power of 1.
Understanding exponentiation is crucial for converting logarithmic equations into exponential form. It bridges the gap between the input (base and exponent) and output (result), helping students see the relationship between numbers and their powers.
Base of a Logarithm
The base of a logarithm is an essential part of both logarithmic and exponential equations. It defines the number that must be raised to a certain power.
  • The base tells us how many times the multiplication must take place.
  • In \( \log_{b} a = c \), the base \( b \) determines the framework of the logarithmic and exponential relationship.
  • For instance, with \( \log_{10} 10 = 1 \), the base is 10, indicating that 10 should be raised to what power (in this case, 1) to result in 10.
Grasping the base of a logarithm is vital for switching between logarithmic and exponential forms. As the axis of these operations, understanding the base allows students to simplify, solve, and interpret mathematical expressions with confidence.

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Most popular questions from this chapter

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

Graph each function by finding ordered pair solutions, plotting the solutions, and then drawing a smooth curve through the plotted points. $$ f(x)=e^{x}-3 $$

Write each as a logarithmic equation. See Example 2. $$ 10^{-2}=\frac{1}{100} $$

Write each as a logarithmic equation. See Example 2. $$ 5^{1 / 2}=\sqrt{5} $$

Graph each function by finding ordered pair solutions, plotting the solutions, and then drawing a smooth curve through the plotted points. $$ f(x)=e^{x+4} $$
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