Problem 27 - Algebra and Trigonometry

Question

\(25-32\) Use the definition of the logarithmic function to find \(x\). $$ \begin{array}{ll}{\text { (a) } \log _{3} 243=x} & {\text { (b) } \log _{3} x=3}\end{array} $$

Step-by-Step Solution

Answer

For (a), \(x = 5\). For (b), \(x = 27\).

Step 1: Understand the Definition of Logarithms

The logarithmic function \(\log_b(a) = x\) is the exponent \(x\) to which the base \(b\) must be raised to obtain the number \(a\). This implies that \(b^x = a\).

Step 2: Solve Part (a) \(\log_{3} 243 = x\)

According to the definition, \(3^x = 243\). We need to find \(x\) such that \(3^x = 243\).- Recognize that \(243 = 3^5\) (since \(3 \times 3 \times 3 \times 3 \times 3 = 243\)).- Therefore, \(x = 5\).

Step 3: Solve Part (b) \(\log_{3} x = 3\)

Here, we're given the equation \(\log_{3} x = 3\), which translates to finding \(x\) such that \(3^3 = x\).- Calculate \(3^3 = 3 \times 3 \times 3 = 27\).- Thus, \(x = 27\).

Key Concepts

Exponents

Exponents are a fundamental concept in mathematics, especially when dealing with logarithms. An exponent is a way of expressing repeated multiplication of the same number. When we write something like \(b^x\), we're saying that the base \(b\) is multiplied by itself \(x\) times. For example, \(3^4 = 3 \times 3 \times 3 \times 3 = 81\).
Understanding exponents is essential when working with logarithms because logarithms are essentially the reverse operation of exponentiation.
You can think of a logarithm as answering the question: "To what power must the base be raised, to yield a particular number?" Exponential functions can grow very fast due to this repeated multiplication aspect, which is why knowing how to handle them can be very useful when solving algebraic equations.

Base of Logarithm

The base of a logarithm is a crucial element that determines the nature of the logarithmic function. When you see something like \(\log_b(a)\), the \(b\) is the base. It tells you what number is being raised to the power. For instance, in \(\log_3(243)\), the base is 3, and you are trying to find the power to which 3 must be raised to get 243.
Different bases can be used, but common ones include base 10 (common logarithm) and base \(e\) (natural logarithm).
The base of the logarithm dictates the scale of the logarithmic function, just like how different units measure the same quantity on different scales. Hence, always pay attention to the base, as it lays the foundation for determining the value of the logarithm.

Solving Logarithmic Equations

Solving logarithmic equations involves finding the unknown variable that makes the equation true. You often start with an equation in form \(\log_b(x) = c\), where you need to figure out the value of \(x\).
The strategy is straightforward due to the definition of logarithms: convert the logarithmic form into exponential form \(b^c = x\). This approach helps bridge the gap between the logarithm and the exponent.
Let's take brain-teaser (a) \(\log_3(243) = x\). This equation translates to finding \(x\) in \(3^x = 243\). By noticing that \(243\) equates to \(3^5\), we find \(x = 5\). A similar tactic can be applied to solve (b) where \(\log_3(x)=3\), turning it into \(3^3 = x\) and solving for \(x = 27\).
Remember, every logarithmic problem can be effectively tackled using the property of reversing it into an exponential context, thus solving for the unknown systematically.