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Chapter 3: Problem 16

Write an equivalent logarithmic equation. $$Q^{n}=T$$

Short Answer

Expert verified
\[ \log_Q(T) = n\]

Step by step solution

01

- Identify the form of the equation

The given equation is in the form of an exponential equation: \[Q^n = T\]
02

- Apply the definition of a logarithm

Using the definition of a logarithm, which states that if \[a^b = c\] then the equivalent logarithmic form is \[ \log_a(c) = b\].
03

- Substitute the values

Substitute the values from the given exponential equation into the logarithmic form:\[ a = Q, \ b = n, \ c = T \]This gives the equivalent logarithmic equation: \[ \log_Q(T) = n\]

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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 an equation where the variable appears in an exponent. In other words, the equation involves a base raised to a power. For example, if you see something like \(Q^n = T\). Here:
  • \(Q\) is the base
  • \(n\) is the exponent or power
  • \(T\) is the result or the value the base raised to the power equals
Exponential equations appear frequently in real-world applications, such as population growth, radioactive decay, and compound interest calculations. Identifying the base, exponent, and result is the first step in working with these kinds of equations.
Logarithm Definition
A logarithm is the inverse function to exponentiation, meaning it helps you find the exponent as opposed to the result. Essentially, if you have an equation like \(a^b = c\), the logarithmic form of this equation is \( \log_a(c) = b \). This helps us work backwards to find the exponent when we know the base and the result. In logarithmic terms:
  • \(a\) is the base
  • \(c\) is the argument or the result
  • \(b\) is the exponent or the logarithm
This process lets you solve for the exponent when the equation is in its exponential form. Knowing the definition of logarithms is crucial for understanding and solving logarithmic equations.
Converting Between Forms
Converting between exponential and logarithmic forms is essential in solving logarithmic equations. For instance, given the exponential equation \(Q^n = T\), you can convert it to its logarithmic form using these steps:
  • First, identify the base \(Q\), the exponent \(n\), and the result \(T\).
  • Apply the logarithm definition: if \(a^b = c\), the logarithmic form is \( \log_a(c) = b \).
  • Substitute the identified values into the logarithmic form: Here, \(a = Q\), \(c = T\), and \(b = n\). This gives us \( \log_Q(T) = n \).
Understanding how to switch views between these forms simplifies complex equations and aids in practical problem-solving, like calculating interest rates or determining the half-life of substances.

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