Problem 6 - Algebra and Trigonometry

Question

\(3-8\) Express the equation in exponential form. $$ \begin{array}{ll}{\text { (a) } \log _{3} 81=4} & {\text { (b) } \log _{8} 4=\frac{2}{3}}\end{array} $$

Step-by-Step Solution

Answer

(a) \(3^4 = 81\), (b) \(8^{2/3} = 4\).

Step 1: Understanding Logarithms

A logarithm \(\log_b (n) = x\) is equivalent to saying that the base \(b\) raised to the power \(x\) gives \(n\). Thus, the logarithmic equation \(\log_b n = x\) can be rewritten as an exponential equation \(b^x = n\).

Step 2: Convert (a) to Exponential Form

Given \(\log_3 81 = 4\), using the understanding from Step 1, we have \(3^4 = 81\). This is the exponential form of the equation.

Step 3: Convert (b) to Exponential Form

Given \(\log_8 4 = \frac{2}{3}\), we can write this as \(8^{\frac{2}{3}} = 4\). This translates the logarithmic equation into exponential format.

Key Concepts

Understanding Logarithmic Equations

Logarithmic equations might seem complex, but they are simply equations involving logarithms that need to be solved for an unknown variable. A logarithmic equation generally takes the form \( \log_b(n) = x \), where the base \( b \), the argument \( n \), and the result \( x \) are connected through exponentiation.
To better understand, let's break it down:

• "Logarithm of \( n \) to the base \( b \)" is asking "to what power do we raise \( b \) to get \( n \)?"
• Solving a logarithmic equation implies converting it to an exponential form for clarity and to find the solution easily.

Let's see this in action with the example of \( \log_3 81 = 4 \). We ask: "What power do we raise 3 to get 81?" The answer is 4. This can be rewritten in exponential form as \( 3^4 = 81 \). Such steps are key in solving equations involving logs, converting them to a more familiar form for computation.

Concept of Logarithms

Logarithms are essential in multiple fields, including mathematics, science, and engineering. They help us understand relationships and solve problems involving exponential growth or decay.
Here are the core elements of logarithms:

• Base: The base is the number that is raised to a power. In \( \log_b(n) = x \), \( b \) is the base.
• Argument: The argument \( n \) is the number for which we're trying to find the power of the base.
• Result: The result \( x \) shows the power to which the base must be raised to obtain the argument.

For example, in the expression \( \log_8 4 = \frac{2}{3} \), the base is 8, the argument is 4, and the result is \( \frac{2}{3} \) because 8 raised to the power of \( \frac{2}{3} \) equals 4.
Understanding these elements helps in solving logarithmic equations and mastering the conversion between different forms in mathematics.

Base and Exponent Relationship

In the world of mathematics, the relationship between base and exponent is significant as it serves as the foundation for understanding not only logarithms but also exponential growth and decay.
Here's what you need to know:

• The base is the number that is multiplied by itself a certain number of times.
• The exponent or power indicates how many times the base is used as a factor in the multiplication.

For example, in \( 3^4 = 81 \), 3 is the base, and 4 is the exponent. The exponential form means multiplying 3 by itself four times: \( 3 \times 3 \times 3 \times 3 = 81 \).
In turn, when converting from a logarithmic to an exponential form as seen in \( \log_8 4 = \frac{2}{3} \), the base 8 raised to the exponent \( \frac{2}{3} \) equals 4. Recognizing this connection provides clarity and simplifies computations, transforming complex problems into workable solutions.