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Chapter 13: Problem 17

Write in logarithmic form. $$9^{2}=81$$

Short Answer

Expert verified
The logarithmic form of the given exponential equation \(9^2 = 81\) is \(\log_9{81} = 2\).

Step by step solution

01

Identify 'a', 'b', and 'x'

In the given exponential equation, \(9^2 = 81\), we have: - the base 'a' as 9 - the exponent 'x' as 2 - the result 'b' as 81
02

Apply the definition of logarithm

Using the definition of logarithm \(\log_a{b} = x\), we can rewrite the given equation. We substitute a = 9, b = 81, and x = 2.
03

Write the equation in logarithmic form

After substituting the values, we get the logarithmic form: \(\log_9{81} = 2\) So, the logarithmic form of the given exponential equation \(9^2 = 81\) is \(\log_9{81} = 2\).

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Key Concepts

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

Exponential Equations
Exponential equations are mathematics' way of expressing how many times a certain number, called the base, is multiplied by itself to reach another number. In the equation \(9^2 = 81\), the base is 9, and the exponent, which tells us how many times to multiply the base by itself, is 2. This means \(9 \times 9 = 81\). Exponential equations are very powerful for quick calculations involving repeated multiplication.
They are commonly found in scientific calculations and growth models like population growth and compound interest.
  • They involve a base (the number that is multiplied).
  • The exponent determines how many times the base is used in the multiplication.
  • The result represents the outcome of the exponential operation, which in our example is 81.
Logarithms
Logarithms are the flipside of exponential equations. They help us find out how many times we need to use the base as a factor to reach a certain number. Using logarithms, we can convert complex multiplication tasks into simpler addition tasks. Consider our example \(9^2 = 81\). In logarithmic form, this becomes \(\log_9{81} = 2\). Here, the base of 9, when raised to the power of 2, gives you 81. Writing and understanding exponential equations as logarithms is useful, especially in computation and problem-solving.
They provide a streamlined way to handle large numbers and simplify problem-solving by dealing with multiplicative aspects in terms of addition.
  • Logarithms are the inverse operation of exponentiation.
  • They can help solve equations where the exponent is unknown.
  • They are vital in fields requiring scientific computation and data science.
Base and Exponent
The concepts of base and exponent are central to understanding both exponential equations and logarithms. The base is the number that is multiplied, while the exponent tells you how many times to use the base in a multiplication. In \(9^2 = 81\), the base is 9, and the exponent is 2, indicating the base is multiplied by itself two times. When converted to logarithmic form \(\log_9{81} = 2\), 9 remains the base, while 2 represents the number of times 9 multiplies with itself to become 81.
The understanding of base and exponent is not only fundamental to simplify calculations but also to solve complex equations efficiently.
  • The base acts as the repeated factor in an exponential expression.
  • The exponent indicates the number of times the base appears in the multiplication.
  • A solid grasp of these concepts is crucial for studying higher mathematics like calculus and algebra.

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Most popular questions from this chapter

Write as a single logarithm. Assume the variables are defined so that the variable expressions are positive and so that the bases are positive real numbers not equal to \(1 .\) $$\log _{4} 7+\log _{4} x$$

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Do all functions have inverses? Explain your answer.
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