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Chapter 9: Problem 69

Simplify. $$\log _{t} t^{10}$$

Short Answer

Expert verified
\(\text{log}_t (t^{10}) = 10\)

Step by step solution

01

Understand the Logarithmic Identity

Recall the logarithmic identity \(\text{log}_b (b^x) = x \). This tells us that the logarithm of a number with the same base as the logarithm will be equal to the exponent.
02

Apply the Logarithmic Identity

Here, the base of the logarithm is \(t\) and the argument is \(t^{10}\). Substitute these values into the identity: \(\text{log}_t (t^{10}) = 10\).

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Key Concepts

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

simplifying logarithms
Simplifying logarithms can make complex equations much easier to work with. Let's start with the basic logarithmic identity \( \text{log}_b (b^x) = x \). This means if you take the logarithm of a power where the base of the logarithm is the same as the base of the power, you end up with the exponent. In our exercise, we apply this identity to \( \text{log}_t (t^{10}) \). By the identity, we see that the result is simply the exponent: \( 10 \). Simplifying logarithms typically involves recognizing identities and properties of logarithmic expressions. Once you understand these properties, you can quickly simplify even the most complicated-looking logarithms.
exponents and logarithms
Exponents and logarithms are closely related concepts. An exponent indicates how many times a number, called the base, is multiplied by itself. For example, in \( t^{10} \), \( 10 \) is the exponent and \( t \) is the base. Logarithms, on the other hand, answer the question, 'To what exponent must we raise the base to get a certain number?' Thus, \( \text{log}_t (t^{10}) \) asks, 'To what exponent must we raise \( t \) to get \( t^{10} \)?' The answer is clearly \( 10 \), because \( t \) raised to the power of \( 10 \) is indeed \( t^{10} \). Understanding this relationship helps to easily move back and forth between exponential and logarithmic forms.
logarithmic functions
Logarithmic functions are the inverse of exponential functions. While an exponential function grows rapidly, a logarithmic function grows slowly. The general form of a logarithmic function is \( \text{log}_b (x) \), where \( b \) is the base and \( x \) is the argument. In our exercise, the function \( \text{log}_t (t^{10}) \) illustrates this relationship. Unveiling these functions' core is crucial for solving various mathematical problems, from basic algebra to advanced calculus. These functions also share important properties such as:
  • \textbf{Product Property}: \( \text{log}_b (xy) = \text{log}_b (x) + \text{log}_b (y) \)
  • \textbf{Quotient Property}: \( \text{log}_b (x/y) = \text{log}_b (x) - \text{log}_b (y) \)
  • \textbf{Power Property}: \( \text{log}_b (x^k) = k \text{log}_b (x) \)
By understanding these properties, you can manipulate and simplify logarithmic expressions more easily.

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Most popular questions from this chapter

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