Problem 42 \(f(x)=\log _{3}(x-1)\)... [FREE SOLUTION]

Chapter 5: Problem 42

\(f(x)=\log _{3}(x-1)\)

Short Answer

Expert verified

The logarithm function \(f(x) = \log _{3}(x-1)\) is defined as the power to which the base (here, 3) must be raised to get \(x-1\). By substituting \(x = 4\), we get \(f(4) = 1\), and by substituting \(x = 10\), we get \(f(10) = 2\).

Step by step solution

Define the Function

Here, the function given is \(f(x) = \log _{3}(x-1)\). The logarithm function is defined as \(log_{b}(a) = c\), meaning the base \(b\) raised to the power \(c\) equals \(a\). In this case, the base is \(3\). We substitute the values of \(x\) into the function to find the function value.

Substitute the values

Let's substitute \(x = 4\) as an example. Now the function will be \(f(4)=\log _{3}(4-1) = \log _{3}(3)\). According to the log rule, the logarithm of any number to the same base is 1. So, \(f(4) = 1\).

Evaluate for different values

As another example, let's substitute \(x = 10\). This results in \(f(10) = \log_{3}(10-1) = \log_{3}(9)\). Using the rules of logarithms, this evaluates to \(f(10) = 2\), since \(3^2 = 9\). Thus, by substituting appropriate values of \(x\) into the function, we can find the respective function values.

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Key Concepts

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

Base of Logarithm

A logarithmic function is based on the concept of exponents. When a logarithm appears, it includes a base which tells you the power you need to raise a specific number to get another number. In the function given, \(f(x) = \log_{3}(x-1)\), the base of the logarithm is \(3\). This means when you compute a logarithm, you're essentially asking, "3 raised to what power gives me the number I'm looking for?"

If the base is unknowable, the logarithm cannot be calculated directly.
The base can often appear in real-world scenarios, such as in scientific measures of growth or decay rates, where a specific reference point is key.

The common bases used are \(10\) (common logarithm) and \(e\) (natural logarithm), but here, our base \(3\) specifically tells us to use powers of \(3\). Therefore, understanding the base allows straightforward evaluation of any logarithmic function.

Evaluation of Logarithms

To evaluate a logarithmic function is to find its numerical value for given inputs. Evaluate simply means "solve." Look at the function \(f(x) = \log_{3}(x-1)\). To find the function's value for \(x = 4\), substitute \(x\) into the function, transforming it into \(\log_{3}(4-1)\), which simplifies to \(\log_{3}(3)\).

The fundamental property here is that \(\log_{b}(b) = 1\); thus, \(\log_{3}(3) = 1\).
Another simple rule is \(\log_{b}(1) = 0\) because any number raised to the power of zero is one.

Consider another instance with \(x = 10\), leading to \(\log_{3}(10-1) = \log_{3}(9)\). Recognizing that \(3^2 = 9\) makes \(\log_{3}(9) = 2\). Each time we evaluate, it's valuable to know or derive the relevant powers of the base. This skill largely assists in simplifying our calculations and providing a clear path to the correct solution.

Logarithmic Rules

Logarithmic functions follow many handy rules that stem from their relationship with exponents. Let's quickly break down some essential rules that will make working with any logarithmic function smoother.

**Product Rule**: \(\log_{b}(mn) = \log_{b}(m) + \log_{b}(n)\). Use this when you multiply numbers inside a log.
**Quotient Rule**: \(\log_{b}\left(\frac{m}{n}\right) = \log_{b}(m) - \log_{b}(n)\). This is handy when dividing.
**Power Rule**: \(\log_{b}(m^n) = n \cdot \log_{b}(m)\). If you encounter exponents inside a log expression, this is your go-to property.

These rules not only help in evaluating expressions but also in simplifying complex equational setups. They provide the flexibility to rewrite logs in multiple forms, ensuring computations become less cumbersome and more intuitive. The exercise's example shows them in action when converting \(\log_{3}(9)\) to \(2\) by realizing \(3^2 = 9\). Efficiently recognizing these properties supports both quick calculations and deeper conceptual clarity in logarithmic functions.

91影视

\(f(x)=\log _{3}(x-1)\)

Short Answer

Step by step solution

Define the Function

Substitute the values

Evaluate for different values

Key Concepts

Base of Logarithm

Evaluation of Logarithms

Logarithmic Rules

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