Problem 5 - College Algebra

Question

Write each equation in its equivalent exponential form. $$ 5=\log _{b} 32 $$

Step-by-Step Solution

Answer

The equivalent exponential form of the given logarithmic equation is \(b^5 = 32\).

Step 1: Identify components

From the given equation \(5 = \log_b{32}\), we can identify the following: Base (b), Exponent (5), and Result (32).

Step 2: Convert to exponential form

We convert the logarithmic equation into its equivalent exponential form based on the key concept \( \log_b{a} = c \) is the same as \( b^c = a \). So we have: \(b^5 = 32\).

Key Concepts

Logarithmic Equations

Logarithmic equations are fascinating mathematical expressions where the logarithm of a number is set equal to another value. In our given exercise, we see it as \(5 = \log_b{32}\). A logarithm helps in solving for the exponent in an exponential equation. It answers the question: "To what power must the base be raised, to yield a certain number?"
For example:

• The given logarithm \(\log_b{32}\) asks, "What power should \(b\) be raised to, to get \(32\)?"

Understanding logarithmic equations is crucial as they are the inverses of exponential equations and play a vital role in many fields such as science, engineering, and finance.

Base and Exponent

The base and exponent are two pivotal elements in both exponential and logarithmic equations. In our example \(5 = \log_b{32}\):

• \(b\) represents the base. It is the number that is being multiplied.
• 5, which is the output of the log, represents the exponent. It tells us how many times the base is multiplied by itself to obtain a result (here, 32).

In the fundamental concept of exponentiation \(b^c = a\), \(b\) is called the base, \(c\) is the exponent, and \(a\) is the result.
Understanding the relationship between base and exponent helps in effortlessly converting logarithmic equations to exponential forms and vice-versa.

Converting Between Forms

Converting logarithmic equations to exponential form is a straightforward process that makes solving such equations easier. The key concept for converting is understanding that \(\log_b{a} = c\) translates to \(b^c = a\).
In our example, where \(5 = \log_b{32}\), we have:

• The base \(b\) remains the same.
• The exponent \(5\) now becomes the power to which the base is raised.
• The result \(32\) is now written as the outcome of the exponential form.

Thus, the equivalent exponential form is \(b^5 = 32\).
By converting between logarithmic and exponential forms, it becomes easier to visualize and solve problems that involve growth, decay, and many other real-world phenomena. This ability offers an intuitive grasp on how quantities scale with exponentiation.