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Chapter 5: Problem 64

Condense the expression to the logarithm of a single quantity. \(\log _{5} 8-\log _{5} t\)

Short Answer

Expert verified
The logarithm of a single quantity for the expression \(\log _{5} 8-\log _{5} t\) is \(\log _{5} \frac{8}{t}\).

Step by step solution

01

Recognizing the Logarithmic Identity

The problem involves a subtraction between two log expressions with the same base, which directly correlates with the following identity: \(\log_a \frac{m}{n} = \log_a m - \log_a n\). Hence, the subtraction can be rewritten as a single logarithmic expression where subtraction changes into division.
02

Apply the Logarithmic Identity

Replacing the original equation \(\log _{5} 8-\log _{5} t\) with the identity, it becomes \(\log _{5} \frac{8}{t}\).

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Key Concepts

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

Properties of Logarithms
Logarithms have unique properties that make them powerful for simplifying mathematical expressions. These properties allow us to combine or separate logarithmic expressions easily. Knowing these can help when dealing with larger problems involving exponential and logarithmic functions. Here are some essential properties:
  • Product Property: \( \log_a(mn) = \log_a m + \log_a n \)
  • Quotient Property: \( \log_a \left(\frac{m}{n}\right) = \log_a m - \log_a n \)
  • Power Property: \( \log_a(m^n) = n \cdot \log_a m \)
These properties are derived from the basics of logarithms and serve as identities that help in solving various types of logarithmic equations. They are particularly useful for transforming complicated expressions into simplified forms.
Condensing Logarithms
Condensing logarithms refers to the process of combining several logarithmic expressions into a single one. This is often done by using the logarithmic identities mentioned earlier. The aim is to express multiple log terms as a single logarithm.
In the context of the given problem, when you have something like \( \log_a m - \log_a n \), you can condense it using the quotient property to form \( \log_a \left(\frac{m}{n}\right) \).
This transformation is useful not just for simplicity, but also for solving equations, as working with a single log term makes it easier to analyze and manipulate mathematically.
Logarithm Subtraction
Logarithm subtraction occurs when you have one logarithm being subtracted from another with the same base. If you see something like \( \log_a x - \log_a y \), you are dealing with logarithmic subtraction. The beauty of this action comes to light when you apply the quotient property.
  • By using the rule \( \log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y \), you can condense this into a single logarithm by transforming the subtraction into a division inside the log.
This property ensures that complex expressions become more manageable through simplification, assisting in solving or analyzing further math problems efficiently.
Division in Logarithms
The division appears in logarithms when you transform subtraction into a division using the quotient rule. This is a key feature of working with logarithmic expressions.
Let's take the example from the problem:
  • \( \log_5 8 - \log_5 t \) becomes \( \log_5 \left(\frac{8}{t}\right) \)
Here, the division inside the logarithmic function replaces subtraction. This transformation makes it easier to work with fewer logarithmic entities. It reduces computational complexity and can help in the rearrangement of equations where only one logarithm is desired or necessary.

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Most popular questions from this chapter

In Exercises 79-88, sketch the graph of the equation. $$ y=\frac{4}{1-3 x} $$

The graph of \(f(x)=\log _{3} x\) contains the point \((27,3)\).

In Exercises 89-92, graph the exponential function. $$ f(x)=-3^{x}+4 $$

\(x=e^{-2 / 3}\)

\(e^{2}=7.3890 \ldots\)
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