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Chapter 3: Problem 19

Write each equation in its equivalent logarithmic form. $$7^{y}=200$$

Short Answer

Expert verified
The logarithmic form of the given exponential equation \(7^{y} = 200\) is \( \log_7 200 = y \)

Step by step solution

01

Identify the base, exponent and result of the exponential equation

In the given equation \(7^{y} = 200\), the base 'a' is 7, the exponent 'b' is 'y', and the result 'c' is 200.
02

Write the equation in logarithmic form

Using the formula described in the analysis, the equation can be written as \( \log_7 200 = y \)

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Key Concepts

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

Exponential Equations
An exponential equation is a mathematical expression in which a variable appears as an exponent. These equations can be used to model situations involving growth or decay, such as population growth or radioactive decay.
The general form of an exponential equation is:
  • Base\(^\text{Exponent} = ext{Result}\)
In the original exercise, the equation provided is an exponential equation:
  • \(7^y = 200\)
This indicates that when base 7 is raised to the power of \( y \), the result is 200. Learning to arrange and solve exponential equations helps in converting them to more manageable forms, such as logarithmic equations, for easy manipulation and solving.
Logarithms
Logarithms are the inverse operations to exponentiation. They are used to determine the exponent that a base number is raised to in order to yield a certain value.
The basic definition of a logarithm is:
  • If \( a^b = c \), then \( \log_a c = b\)
In logarithmic terms, it answers the question: "To what power must the base be raised to produce this number?" In the equivalent logarithmic form of the exercise, \( \log_7 200 = y \), we find the power \( y \) that 7 must be raised to in order to result in 200.
Base and Exponent Identification
Identifying the base and exponent in an equation is crucial when converting it from exponential to logarithmic form. This concept highlights how parts of an equation relate and transform between forms.
For example, in the equation \(7^y = 200\):
  • Base: The base is 7, which is the repeated factor in the power.
  • Exponent: The exponent is \( y \), representing how many times the base is used as a factor.
  • Result: The result of this expression when evaluated, which is 200 in the equation.
Understanding these components helps us to rewrite exponential equations as logarithmic equations easily and vice-versa.
Equivalent Equations
Equivalent equations are expressions that represent the same relationship or value, though they may look different. They are essential in mathematics for showing that different forms of equations can express the same principles and results.
When transforming an exponential equation into its logarithmic form, you're seeking its equivalent expression. This doesn鈥檛 change the underlying relationship, just the way it's expressed.For \(7^y = 200\) and \(\log_7 200 = y\) are equivalent equations. Although they appear different, they describe the same mathematical truth:
  • An exponent and its corresponding logarithm convey the same information about numbers involved.
Recognizing and writing equivalent equations strengthens problem-solving capabilities and mathematical understanding.

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Most popular questions from this chapter

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Solve each equation in Exercises \(146-148 .\) Check each proposed solution by direct substitution or with a graphing utility. $$(\log x)(2 \log x+1)=6$$

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