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Chapter 4: Problem 34

Write the equation in logarithmic form. $$ M^{3}=N $$

Short Answer

Expert verified
\(\log_M(N) = 3\)

Step by step solution

01

- Understand the Components of the Exponential Form

The given equation is in the exponential form: \[M^{3}=N\]Here, M is the base, 3 is the exponent, and N is the result.
02

- Apply the Definition of Logarithms

To convert an exponential equation to logarithmic form, use the definition of logarithms. The logarithmic form of the equation \(a^b=c\) is \(\log_a(c) = b\).
03

- Write the Equation in Logarithmic Form

Using the definition from Step 2, the given equation \(M^3 = N\) is written in logarithmic form as: \[\log_M(N) = 3\]

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Key Concepts

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

Understanding Exponential Equations
An exponential equation is a mathematical expression where a constant base is raised to a variable exponent. In our exercise, the given exponential equation is represented as \(M^3 = N\). Here:
  • \(M\) is the base
  • \(3\) is the exponent
  • \(N\) is the result
This means we multiply the base \(M\) by itself 3 times to get \(N\).
An example of simpler exponential equations could be \(2^3 = 8\), where 2 is the base, raising it to the power of 3 results in 8.
To solve any problems involving these equations, you need to understand the relationship between the base, the exponent, and the result.
Introduction to Logarithms
Logarithms are the inverse operation of exponentiation. If an exponential equation is in the form \(a^b = c\), then its logarithmic form is \( \log_a(c) = b\).
In logarithms:
  • The base (\(a\)) of the logarithm is the same as the base of the exponential form.
  • The result of the logarithmic function (\(c\)) is the exponent (\(b\)) in the exponential form.
  • The argument of the logarithm (\(c\)) is the same as the result of the exponential equation.
So, in our exercise, \(M^3 = N\) converts to its logarithmic form as \(\text \log_M(N) = 3\). This reads as 'logarithm base \(M\) of \(N\) equals 3'. Logarithms help simplify equations where solving for the exponent is necessary. They are a crucial tool in algebra and calculus for dealing with exponential growth and decay, among other applications.
Mathematical Notation
Mathematical notation is the language used to express mathematical ideas and arguments. For exponential equations, we use the notation \( \text a^b = c \text\), where:
  • \(a\) is the base
  • \(b\) is the exponent
  • \(c\) is the result
For logarithmic form, we use \( \log_a(c) = b\). In this notation:
  • \(\text \log\text \) signifies the logarithm
  • \(a\) inside the parentheses is the base of the logarithm
  • \(c\) is the argument (the number you are taking the logarithm of)
  • \(b\) is the result
This standardized notation allows mathematicians to communicate complex ideas clearly and efficiently. Understanding and correctly using mathematical notation is essential for solving problems accurately and effectively.

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Most popular questions from this chapter

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. \(\log _{7}(12-t)=\log _{7}(t+6)\)

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. \(\log (p+17)=4.1\)

Solve for the indicated variable. \(A=P(1+r)^{t}\) for \(t\) (used in finance)

Solve the equation. Write the solution set with the exact values given in terms of common or natural logarithms. Also give approximate solutions to 4 decimal places. \(e^{2 x}=-7 e^{x}\)

Solve the equation. Write the solution set with the exact values given in terms of common or natural logarithms. Also give approximate solutions to 4 decimal places. \(80=320 e^{-0.5 t}\)
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