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Chapter 3: Problem 7

write each equation in its equivalent exponential form. $$\log _{6} 216=y$$

Short Answer

Expert verified
The equivalent exponential form of the given equation is \(6^y = 216\)

Step by step solution

01

Review the Equation

Identify the base, exponent and result in the equation \(\log _{6} 216=y\). Note that in this equation: Base = 6, y is the exponent and the result is 216
02

Apply the Exponential Form

Convert the logarithmic equation to the corresponding exponential form by using the base and the exponent. The exponential form of the equation is \(6^y = 216\)

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Key Concepts

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

Exponential Form
When dealing with logarithmic equations, understanding the exponential form is crucial. The exponential form represents numbers as powers. It's a way to express a number by relating it to a base and an exponent. For instance, in the equation \(a^b = c\), \(a\) is the base, \(b\) is the exponent, and \(c\) is the result.
Exponential form is particularly useful because many real-world phenomena and mathematical patterns involve exponential growth or decay. Recognizing these patterns in a mathematical equation allows you to solve problems more efficiently.
  • **Base**: The repeated factor in multiplication.
  • **Exponent**: Tells how many times to use the base in a multiplication.
  • **Result**: The outcome when the base and exponent interact.
Recognizing this form is key to solving both logarithmic and exponential equations effectively.
Base of Logarithm
The base of a logarithm is the number that serves as the reference point in the expression. In a logarithmic equation \(\log_b (c) = a\), \(b\) is the base.
The base determines the manner in which the logarithmic function grows or shrinks, as it indicates the power to which the base must be raised to achieve the given number.
In our specific exercise \(\log_6 216 = y\), the base is 6. This suggests that 6 is raised to some power \(y\) to result in 216.
  • **Traditional bases**: For instance, base 10 is known as the common logarithm, and base \(e\) is used in natural logarithms.
  • **Choosing bases**: Many logarithms use bases that simplify calculations or cater to specific applications, such as binary (base 2) in computers.
Understanding the base is fundamental to converting logarithmic equations to their exponential counterparts.
Convert Logarithmic to Exponential
Converting a logarithmic equation to its equivalent exponential form is a straightforward process involving recognition of each element鈥檚 role. Given a logarithmic expression such as \(\log_b(c) = a\), you convert it to exponential form as \(b^a = c\).
For our example, \(\log_6 216 = y\), the conversion entails:
  • Recognizing that the base of the logarithm (6) will serve as the base in the exponential form.
  • The right side of the equation (\(y\)) becomes the exponent in the exponential form.
  • The number from the logarithm (216) represents the result of the base raised by the exponent.
By converting to exponential form, \(6^y = 216\), this transformation clarifies the relationship between the base, its power, and its product. It's a vital skill for students needing to manipulate and solve equations across mathematics.

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Most popular questions from this chapter

Describe the product rule for logarithms and give an example.

The \(p H\) scale is used to measure the acidity or alkalinity of a solution. The scale ranges from 0 to \(14 .\) A neutral solution, such as pure water, has a pH of 7. An acid solution has a pH less than 7 and an alkaline solution has a pH greater than 7. The lower the \(p H\) below 7 , the more acidic is the solution. Each whole-number decrease in \(p H\) represents a tenfold increase in acidity. (GRAPH CAN'T COPY). The \(p H\) of a solution is given by $$\mathrm{pH}=-\log x$$ where \(x\) represents the concentration of the hydrogen ions in the solution, in moles per liter. Use the formula to solve. Express answers as powers of \(10 .\) a. The figure indicates that lemon juice has a pH of 2.3. What is the hydrogen ion concentration? b. Stomach acid has a pH that ranges from 1 to 3. What is the hydrogen ion concentration of the most acidic stomach? c. How many times greater is the hydrogen ion concentration of the acidic stomach in part (b) than the lemon juice in part (a)?

will help you prepare for the material covered in the next section. In each exercise, evaluate the indicated logarithmic expressions a. Evaluate: \(\log _{3} 81\) b. Evaluate: \(2 \log _{3} 9\) c. What can you conclude about \(\log _{3} 81,\) or \(\log _{3} 9^{2} ?\)

graph f and g in the same viewing rectangle. Then describe the relationship of the graph of g to the graph of f. $$f(x)=\ln x, g(x)=\ln (x+3)$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I can use any positive number other than 1 in the changeof-base property, but the only practical bases are 10 and \(e\) because my calculator gives logarithms for these two bases.
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