Problem 69 - Algebra 2 and Trigonometry

Question

If \(\mathrm{f}(x)=\log _{3} x,\) find \(\mathrm{f}(81)\)

Step-by-Step Solution

Answer

\(f(81) = 4\) because 81 is \(3^4\).

Step 1: Understand the Function

The function given is \(f(x) = \log_{3} x\), which represents the logarithm of \(x\) with base 3. This means we are looking for the exponent to which 3 must be raised to get \(x\).

Step 2: Identify Input Value

We need to find \(f(81)\). This means we are looking for the value of the logarithm: \(\log_{3} 81\).

Step 3: Express 81 as a Power of 3

Since we want \(\log_{3} 81\), it's helpful to express 81 in terms of a power of 3. Notice that \(81 = 3^4\), because \(3 \times 3 \times 3 \times 3 = 81\).

Step 4: Apply Logarithm Properties

Using the properties of logarithms, specifically \(\log_{b}(b^k) = k\), we have \(\log_{3}(3^4) = 4\). So, \(f(81) = 4\).

Key Concepts

Logarithm Properties

Logarithms are mathematical operations that help us to deal with exponential equations. They have certain properties that make calculations a lot easier. The key property used in solving \(f(x) = \log_{3} x\) for \(f(81)\) is the change of a power into a multiplication. In general, the property \(\log_{b}(b^k) = k\) says that when you have the logarithm of an exponent where the base of the exponent matches the base of the logarithm, the result is simply the exponent itself.

This property helps us simplify expressions and solve problems efficiently. For example, since \(81 = 3^4\), it follows that \(\log_{3}(3^4) = 4\).

Another useful property is the product rule: \(\log_{b}(xy) = \log_{b}(x) + \log_{b}(y)\). However, in this exercise, the decomposition of 81 into a power of 3 avoids the need to use the product rule.

Exponents

Exponents are a way of representing repeated multiplication. For example, \(3^4\) is shorthand for \(3 \times 3 \times 3 \times 3\). In the function \(f(x) = \log_{3} x\), figuring out \(f(81)\) requires expressing 81 as a power of 3.

To break it down, you calculate powers step-by-step:

• \(3^1 = 3\)
• \(3^2 = 9\)
• \(3^3 = 27\)
• \(3^4 = 81\)

Here, we see that \(3^4 = 81\), which matches the input value for the function. Understanding exponents lets us handle problems involving powers directly and converts logarithmic concepts into more accessible forms.

Logarithmic Base Conversion

Sometimes, you might want to change the base of a logarithm to simplify calculations or comparison. This is known as logarithmic base conversion. Although base conversion wasn't directly used in the exercise for \(f(x) = \log_{3} x\), understanding it is important for broader contexts.

The change of base formula allows you to convert a logarithm from one base to another: \[\log_{b}(x) = \frac{\log_{k}(x)}{\log_{k}(b)}\]\where \(k\) is a new base that you choose, often 10 or \(e\) for convenience in calculations.

Base conversion is particularly helpful when you have a calculator that computes logarithms in only one base (like base 10 or \(e\)) but you need to work with different bases in mathematical problems.