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Chapter 13: Problem 12

Express the given equations in logarithmic form. $$(12)^{0}=1$$

Short Answer

Expert verified
\(\log_{12}(1) = 0\)

Step by step solution

01

Understand the Given Exponential Equation

The given equation is \((12)^0 = 1\). The base is 12, the exponent is 0, and the result is 1. We need to express this in logarithmic form.
02

Structure of Logarithmic Expression

In general, an exponential form \(a^b = c\) can be written in logarithmic form as \(\log_a(c) = b\). This form helps to express the relationship between the base, exponent, and the result as a logarithm.
03

Apply the Logarithmic Structure

Using the structure from the previous step, identify \(a = 12\), \(b = 0\), and \(c = 1\) in the equation \((12)^0 = 1\). Thus, the logarithmic form is \(\log_{12}(1) = 0\).

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Key Concepts

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

Exponential Equations
Exponential equations involve expressions where a number, known as the base, is raised to a power, called the exponent. Consider an equation
  • The base, often denoted as \(a\), is the number being multiplied by itself a certain number of times.
  • The exponent, denoted as \(b\), indicates how many times the base is multiplied.
In the equation \((12)^0 = 1\), 12 is the base, and 0 is the exponent. One interesting property of exponential equations is that any number raised to the power of zero results in one. This is because multiplying any number by itself zero times is conceptually similar to saying "no multiplication is happening," leaving us with the identity element in multiplication: one.
Base and Exponent
The base and exponent are two critical components of exponential expressions that determine the expression's value. Let鈥檚 break these components further:
  • Base (12): This is the starting value that is going to be multiplied. It鈥檚 effectively the number that gives the expression its weight in calculations.
  • Exponent (0): Represents the power to which the base is raised. In simple terms, it鈥檚 the number of times you multiply the base by itself.
In our example, since the exponent is zero, it leads to a special case where any non-zero base raised to the power of zero is always equal to one. This rule is significant because it offers a consistent way to manage powers of bases, ensuring that mathematical equations behave predictably.
Logarithmic Expression
Logarithmic expressions present exponential equations in another format that reveals how many times you need to multiply the base to get a certain number. They follow this general structure: \[\log_a(c) = b\]This reads as "log base \(a\) of \(c\) equals \(b\)." In our problem, the task is to convert \((12)^0 = 1\) into logarithmic form.
  • Base (12): Here, it is referred to as the base of the logarithm.
  • Result (1): The number you achieve by raising the base to the given exponent.
  • Exponent (0): In the logarithmic form, it is the result of the logarithmic function.
So, when we transform \((12)^0 = 1\) into a logarithmic expression, we align it as \(\log_{12}(1) = 0\). This equation asks, "To what power do we need to raise 12 to get 1?" And the answer, just as we've seen, is zero, demonstrating the unique rule that any non-zero number to the power of zero equals one.

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Most popular questions from this chapter

Show that the given functions are inverse functions of each other. Then display the graphs of each function and the line \(y=x\) on a graphing calculator and note that each is the mirror image of the other across \(y=x\) $$y=10^{x / 2} \text { and } y=2 \log _{10} x$$

Use a calculator to evaluate (to three significant digits) the given numbers. $$(2 e)^{-\sqrt{2}}$$

Determine the value of the unknown. $$\log _{10} 10^{0.2}=x$$

Show that the given functions are inverse functions of each other. Then display the graphs of each function and the line \(y=x\) on a graphing calculator and note that each is the mirror image of the other across \(y=x\) $$y=e^{x} \text { and } y=\log _{e} x$$

Express the given equations in exponential form. $$5 \log _{243} 3=1$$
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