Problem 2 - Algebra and Trigonometry

Question

The function \(f(x)=\log _{8} x\) is the logarithm function with base ________, So \(f(9)=\) _________ ,\(f(1)=\) _______,\(f(t)=\) _________, \(f(81)=\) __________, and \(f(3)=\) _________

Step-by-Step Solution

Answer

The base is 8. Exact values: \(f(9)=\log_{8} 9\), \(f(1)=0\), \(f(t)=\log_{8} t\), \(f(81)=\log_{8} 81\), \(f(3)=\log_{8} 3\).

Step 1: Identify the base of the logarithm

The base of the logarithm function is the number indicated after the word "log" and before the variable or number. In this case, it is 8. Therefore, the base of the logarithm function \(f(x) = \log_{8} x\) is 8.

Step 2: Compute \(f(9)\)

To compute \(f(9)\), we evaluate \(\log_{8} 9\). This is the power to which 8 must be raised to obtain 9. Since 9 is not a simple power of 8, this will typically remain as an expression or be approximated. Thus, \(f(9) = \log_{8} 9\).

Step 3: Compute \(f(1)\)

For any base, \(\log_{b} 1 = 0\) because any non-zero number raised to the power of 0 is 1. So, \(f(1) = \log_{8} 1 = 0\).

Step 4: Express \(f(t)\)

To express \(f(t)\) in this context, we use the given logarithm function. Thus, \(f(t) = \log_{8} t\).

Step 5: Compute \(f(81)\)

To compute \(f(81)\), we find \(\log_{8} 81\). Similarly to \(f(9)\), since 81 is not a direct power of 8, it will remain as \(\log_{8} 81\) or be further approximated. Thus, \(f(81) = \log_{8} 81\).

Step 6: Compute \(f(3)\)

For \(f(3)\), we evaluate \(\log_{8} 3\). Since 3 cannot be easily expressed as a power of 8, the result remains \(\log_{8} 3\) as an exact form. Therefore, \(f(3) = \log_{8} 3\).

Key Concepts

Logarithm Base

The logarithm base is a key part of the log function determining how numbers are represented as exponents. In \(f(x) = \log_{8} x\), the base is 8. This base indicates that the log function is asking, "What power must 8 be raised to reach \(x\)?" We write logs with a base that is shown as a subscript.
This concept is important because the properties and evaluation of logs vary with different bases.

• Understanding the base is crucial for interpreting log expressions correctly.
• Bases greater than 1 mean the log is positive when \(x > 1\), zero when \(x = 1\), and negative when \(0 < x < 1\).

Grasping the base of a log is fundamental before moving on to function evaluations or further properties of logarithms.

Function Evaluation

Evaluating a logarithmic function means calculating the log for a specific input value. Using the function \(f(x) = \log_{8} x\), we can evaluate different values:

• For \(f(9)\), we find \(\log_{8} 9\). This means we look for what power 8 needs to become 9.
• For \(f(1)\), since any nonzero number raised to the power of zero is 1, \(\log_{8} 1 = 0\).
• For \(f(t)\), we express it as \(\log_{8} t\), which shows we want the exponent for 8 to equal \(t\).
• For \(f(81)\) and \(f(3)\), simplification might not be easy, so these stay as expressions \(\log_{8} 81\) and \(\log_{8} 3\).

Evaluating log functions can involve keeping expressions exact or computing approximate values using calculators for non-simple cases.

Logarithmic Expression

A logarithmic expression is simply a log function like \(\log_{8} x\). It is a way to express powers and exponents in a specific form. In basic terms:

• The log expression states the problem of finding which power a certain base needs to equal the clculated value.
• Rewriting logarithmic expressions can simplify complex arithmetic involving exponents.
• Each log expression keeps finite information about multiplicative relationships compactly.

Understanding logarithmic expressions helps to manipulate and simplify expressions that are not easily interpreted in exponential form.

Properties of Logarithms

The properties of logarithms make them versatile tools in math. They help us to simplify, solve, and manipulate log expressions. Here are some main properties:

• Product Rule: \( \log_b (xy) = \log_b x + \log_b y \), combining products.
• Quotient Rule: \( \log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y \), dealing with fractions.
• Power Rule: \( \log_b (x^n) = n \cdot \log_b x \), breaking down exponents.
• Change of Base Formula: Change to any desired base using \( \log_b x = \frac{\log_k x}{\log_k b} \).

Using these properties translates complex expressions into simpler calculations or alternative representations, helping us resolve logs in various contexts.