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Chapter 5: Problem 9

Express as a product. $$\log _{b} t^{3}$$

Short Answer

Expert verified
\[\forall \log_b t^3\ = 3 \log_b t\]

Step by step solution

01

Recall the Power Rule of Logarithms

The Power Rule of logarithms states that \(\forall \log_b (a^c) = c \log_b a\). This rule allows us to express the logarithm of a power as a product.
02

Apply the Power Rule

Given the logarithmic expression \(\forall \log_b t^3\), use the Power Rule to rewrite it. This gives us: \[\log_b t^3 = 3 \log_b t\].

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Key Concepts

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

Power Rule of Logarithms
One of the key concepts in logarithmic expressions is the Power Rule of Logarithms. This rule is incredibly useful for simplifying logarithmic expressions.
It states that for any logarithm of a number raised to a power, you can move the exponent to the front as a multiplier. Mathematically, it looks like this:
\( \forall \, \text{log}_b (a^c) = c \, \text{log}_b (a) \).
Let's break it down with an example. If we have \( \text{log}_b (t^3) \),
the Power Rule allows us to rewrite it as:
\( 3 \, \text{log}_b (t) \).
This is super useful because it simplifies a potentially complex logarithmic expression into a more manageable form.
Logarithmic Properties
Logarithms have several important properties that make them useful in mathematics.
Some of the key properties include:
  • Power Rule: As we've discussed, \( \text{log}_b (a^c) = c \, \text{log}_b (a) \).
  • Product Rule: The logarithm of a product is equal to the sum of the logarithms of the factors. This is written as \( \text{log}_b (xy) = \text{log}_b (x) + \text{log}_b (y) \).
  • Quotient Rule: The logarithm of a quotient is the difference between the logarithm of the numerator and the denominator. This property is expressed as \( \text{log}_b (x / y) = \text{log}_b (x) - \text{log}_b (y) \).
  • Change of Base Formula: You can change the base of a logarithm using the formula \( \text{log}_b (a) = \frac{\text{log}_k (a)}{\text{log}_k (b)} \), for any base \( k \).
These properties are helpful for breaking down and simplifying more complicated logarithmic expressions. Remembering these properties will make working with logarithms much easier.
Precalculus
Precalculus is an important course because it lays the foundation for calculus and other advanced mathematical topics. In precalculus, you'll often encounter various types of functions, including logarithmic functions.
Understanding logarithmic properties and rules is essential for success in precalculus. For example, knowing the Power Rule of Logarithms helps simplify complex expressions and solve logarithmic equations.
In addition to logarithms, precalculus covers functions, sequences, series, and trigonometry.
  • Functions: Explore properties of different types of functions like quadratic, polynomial, rational, exponential, and logarithmic functions.
  • Sequences and Series: Understand arithmetic and geometric sequences along with series and their sums.
  • Trigonometry: Study the relationships between the angles and sides of triangles, as well as trigonometric functions.
By mastering these topics, students build a strong mathematical foundation that is crucial for tackling calculus and beyond.

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Most popular questions from this chapter

The following formula can be used to convert Fahrenheit temperatures \(x\) to Celsius temperatures \(T(x):\) $$ T(x)=\frac{5}{9}(x-32) $$ a) Find \(T\left(-13^{\circ}\right)\) and \(T\left(86^{\circ}\right)\) b) Find \(T^{-1}(x)\) and explain what it represents.

Given that \(\log _{a} x=2, \log _{a} y=3,\) and \(\log _{a} z=4\) find $$\log _{a} \frac{\sqrt[4]{y^{2} z^{5}}}{\sqrt[4]{x^{3} z^{-2}}}$$

Solve using any method. $$5^{2 x}-3 \cdot 5^{x}+2=0$$

Determine whether each of the following is true or false. Assume that \(a, x, M,\) and \(N\) are positive. $$\frac{\log _{a} M}{x}=\log _{a} M^{1 / x}$$

Find the \(x\) -intercepts and the zeros of the function. $$h(x)=x^{4}-x^{2}[4.1]$$
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