Problem 19 - Precalculus with Limits

Question

Evaluate the function at the indicated value of \(x\) without using a calculator. \(g(x)=\log _{a} x \quad x=a^{2}\)

Step-by-Step Solution

Answer

\(g(a^2) = 2\)

Step 1: Understanding the Function

The given function is \(g(x) = \log_{a}{x}\). The goal is to evaluate this function at \(x=a^2\).

Step 2: Substitute the Value into the Function

Substitute \(x=a^2\) into the function. This gives \(g(a^2) = \log_{a}{a^2}\).

Step 3: Apply Logarithm Property

Remembering the property of logarithms \(\log_{a}{a^x} = x\), you can simplify \(g(a^2)\) to 2. So, \(g(a^2) = 2\).

Key Concepts

Evaluate Logarithmic Functions

Understanding how to evaluate logarithmic functions is a fundamental skill in algebra that involves finding the value of logarithmic expressions and functions. The process usually requires knowing the base of the logarithm and the value into which you need to substitute.

For instance, when you have a function like \( g(x) = \log_a x \), and you're asked to evaluate it at a specific value of \( x \), say \( x = a^2 \), the first step is to substitute \( x \) with \( a^2 \) in the function, giving you \( g(a^2) = \log_a{a^2} \). By understanding logarithm properties, you can then simplify this expression without needing a calculator.

Logarithm Rules

Logarithms might seem daunting at first, but they follow specific rules that make calculations far more manageable. One of the most vital rules to remember is the 'base and exponent' relationship rule: \( \log_b{b^x} = x \). This property states that if the base of the logarithm and the number inside the logarithm are the same, and the number inside the logarithm is raised to an exponent, the logarithm simplifies to that exponent.

Example:

In our problem \( g(x) = \log_a{a^2} \), using this rule directly gives us the simplicity of \( g(a^2) = 2 \), without further calculation. Other important properties include the product, quotient, and power rules, which help break down more complex logarithmic expressions into solvable parts.

Logarithmic Expressions Without Calculator

Sometimes you need to evaluate logarithmic expressions without the aid of a calculator. This is where your understanding of logarithms and their properties really shines. By internalizing how logarithms work, you know that a logarithm is asking, 'to what exponent must we raise the base to get the number inside the logarithm?'

Whenever the number inside the logarithm is a power of the base, like in the exercise \( g(a^2) = \log_a{a^2} \), you can answer that question without any calculations: the exponent you're looking for is just the exponent of \( a \) itself, which is 2 in this case. It's this logical reasoning, based on the rules of logarithms, that enables you to evaluate logarithmic expressions with confidence, even without a calculator.