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Chapter 4: Problem 16

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

Short Answer

Expert verified
n = \log_Q(T).

Step by step solution

01

Recall the Definition of a Logarithm

The definition of a logarithm states that for any positive real numbers Q and T and any real number n, the equation \(Q^n = T\) is equivalent to \(n = \log_Q(T)\).
02

Apply the Definition

Using the definition from Step 1, rewrite the given equation \(Q^n = T\) as its equivalent logarithmic equation \(n = \log_Q(T)\).

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Key Concepts

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

Logarithm Definition
A logarithm is the inverse operation to exponentiation. It tells us what exponent we need to use, so a specific base yields a given number.
For example, in the equation \(Q^n = T\), the logarithm helps us find out what power \(n\) should be to get \(T\) when starting with \(Q\).
The notation for this is typically written as \(n = \log_Q(T)\) which means 'n is the exponent to which the base Q must be raised, to get T'.

Key points to remember:
  • Logarithms are exponents.
  • The base of the logarithm can't be negative or zero.
  • The argument (the number inside the log) must be positive.
Exponential Form
Exponential form refers to an equation where a number (base) is raised to a power (exponent) to get another number.
In our example, \(Q^n = T\), \(Q\) is the base, \(n\) is the exponent, and \(T\) is the result.
Converting between logarithmic and exponential forms can make solving equations easier.

Here鈥檚 how it works:
  • Start with the exponential form: \(Q^n = T\).
  • Recognize that this is saying 'Q raised to the power of n gives T.'
  • Rewrite in logarithmic form: \(n = \log_Q(T) \) which reads as 'n is the power you raise Q to get T.'
Equivalent Equations
Equivalent equations are different ways of expressing the same relationship between numbers.
For exponential and logarithmic equations, recognizing equivalency can help us solve complex problems.

An example with \(Q^n = T\):
  • The exponential form is \(Q^n = T\).
  • The logarithmic form is \(n = \log_Q(T)\).
  • Both forms describe the same relationship: knowing one allows you to find the other.

    Remember, understanding and converting between these forms helps in solving various mathematical problems more efficiently!

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