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Chapter 9: Problem 9

Write each equation in exponential form. \(\log _{6} 216\)

Short Answer

Expert verified
The exponential form of \(\log_{6} 216\) is \(6^x = 216\).

Step by step solution

01

Understand the Problem

We need to convert the logarithmic expression \(\log_{6} 216\) into its equivalent exponential form. In the expression \(\log_{b} a = c\), \(b\) is the base, \(a\) is the number, and \(c\) is the exponent.
02

Identify Components

From the logarithmic expression \(\log_{6} 216\), identify the base \(b = 6\) and the number \(a = 216\). Assume the expression is equal to some exponent \(c\).
03

Rewrite in Exponential Form

The equivalent exponential form of \(\log_{b} a = c\) is \(b^c = a\). Thus, \(\log_{6} 216\) can be rewritten as \(6^x = 216\), where \(x\) is the unknown exponent.

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Key Concepts

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

Logarithmic Expressions
Logarithmic expressions can be initially confusing, but they follow a consistent pattern. A logarithmic expression \(\log_{b} a\) signifies the power to which the base, \(b\), must be raised to produce the number, \(a\). Understanding this relationship is crucial for converting logarithmic expressions into exponential form. It's helpful to remember:
  • The base of the logarithm tells you what number is being multiplied repeatedly.
  • The number is the result of multiplying the base so many times together.
  • The logarithmic expression yields the power or exponent that shows how many times the base is used in the multiplication.
For example, in \(\log_{6} 216\), the base is 6, and the expression asks "6 raised to what power equals 216?" By turning this into exponential form, we make it easier to solve for the unknown exponent.
Base and Exponent
The concepts of base and exponent are central to understanding exponential form. In exponential expressions, like \(b^c = a\), **b** is the base, **c** is the exponent, and **a** is the result. Breaking down these components can clarify their roles:
  • **Base (b)**: The number that is being multiplied repeatedly.
  • **Exponent (c)**: Indicates the number of times the base is multiplied by itself.
  • **Result (a)**: The outcome of raising the base to the power of the exponent.
In the equation derived from the logarithmic expression, \(6^x = 216\), 6 is the base, and the equation seeks to find **x**, the exponent. Understanding this process is crucial for converting logarithmic equations as it requires recognizing how many times the base must multiply itself to result in the given number.
Exponential Equations
Exponential equations involve expressions where a constant base is raised to an unknown power, equating it to a number. These equations, like \(6^x = 216\), require solving for the exponent, which is the unknown in the equation. Here's how you can tackle them:
  • Recognize both sides of the equation, with the left side being the exponential expression and the right side the result.
  • Aim to express both the base and the result in terms of powers of the same number, if possible. This helps simplify calculations.
  • Where direct calculation is complex, logarithms can be used to find the unknown exponent easily.
In our example, identifying 216 as a power of 6 helps solve for x seamlessly. Breaking down the equation into familiar terms aids in understanding the solution process and demystifying logarithmic to exponential form conversions.

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Most popular questions from this chapter

The number of people \(N\) who will receive a forwarded e-mail can be approximated by \(N=\frac{P}{1+(P-S) e^{-0.35 t}},\) where \(P\) is the total number of people online, \(S\) is the number of people who start the e-mail, and \(t\) is the time in minutes. Suppose four people want to send an e-mail to all those who are online at that time. If there are 156,000 people online, how many people will have received the e-mail after 25 minutes?

Solve each equation or inequality. Round to the nearest ten-thousandth. \(e^{x}>1.6\)

The number of people \(N\) who will receive a forwarded e-mail can be approximated by \(N=\frac{P}{1+(P-S) e^{-0.35 t}},\) where \(P\) is the total number of people online, \(S\) is the number of people who start the e-mail, and \(t\) is the time in minutes. Suppose four people want to send an e-mail to all those who are online at that time. How much time will pass before half of the people will receive the e-mail?

Solve each equation or inequality. Check your solutions. $$ \log _{3}(4 x-5)=5 $$

CHALLENGE Tell whether each statement is true or false. If true, show that it is true. If false, give a counterexample. For all positive numbers \(m, n, x,\) and \(b,\) where \(b \neq 1, n \log _{b} x+m \log _{b} x=\) \((n+m) \log _{b} x\)
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