Problem 7 - College Algebra with Corequisite Support

Question

For the following exercises, rewrite each equation in exponential form. \(\log _{a}(b)=c\)

Step-by-Step Solution

Answer

The exponential form is \(a^c = b\).

Step 1: Understand the Logarithmic Equation

The logarithmic equation given is \(\log_{a}(b) = c\). This means that "\(a\) raised to what power equals \(b\)?" and the answer is \(c\). We aim to convert this into its equivalent exponential form.

Step 2: Convert to Exponential Form

In exponential form, the equation \(\log_{a}(b) = c\) becomes \(a^c = b\). This means that raising the base \(a\) to the power \(c\) results in \(b\).

Key Concepts

Logarithmic Equation

A logarithmic equation provides a way to determine how many times a certain base must be multiplied by itself to reach another number, called the argument. In the expression \( \log_{a}(b) = c \), \(a\) is the base, \(b\) is the argument, and \(c\) is the result or exponent. Essentially, it is solving for the power that transforms a base into a given number. It's like asking, "What power of 10 is 100?" The answer would be 2, since \(10^2 = 100\). This makes logs highly useful in various applications, including fields like engineering and sciences, where they simplify complex multiplicative relationships. Logs reverse the process of exponentiation by telling you how many times to apply the base as a factor to get a certain number.
In understanding these equations, think of logarithms as the inverse operations of exponentials, similar to how subtraction is the inverse of addition.

Base and Power

In mathematics, 'base' usually refers to the number that is raised to a certain power. Within the context of logarithmic and exponential equations, the base is the foundational number upon which the power acts. For example, in the equation \(a^c = b\), \(a\) is the base and \(c\) is the power. The base has a uniquely crucial function because it forms the backbone of exponential calculations. The power, or exponent, tells you how many times the base should be used as a factor.

• If \(c = 2\), then the calculation is \(a \times a\).
• If \(c = 3\), it means \(a \times a \times a\).

These operations highlight the incredible reach of powers, transforming even small bases into significantly large numbers. So, base and power work together to transform seemingly simple numbers into complex calculations with wide-ranging applications, like compounding interest and population growth.

Conversion from Logarithmic to Exponential

Conversion from logarithmic to exponential form is a vital skill that helps bridge the relationship between these two fundamental mathematical concepts. In the equation \( \log_{a}(b) = c \), converting it to exponential form involves rearranging it to \( a^c = b \). This means that the base \(a\) raised to the power \(c\) is equal to \(b\), demonstrating a direct way to understand what logarithms calculate.
To convert, identify the basic components of the logarithmic form:

• \(a\) is the base.
• \(c\) is the power to which the base is raised.
• \(b\) is the result or output.

After identifying these elements, simply place them into the exponential form \(a^c = b\). This transformation not only clarifies the relationship between the numbers but also simplifies calculations, allowing for easier manipulation and understanding. Conversion between these forms is critical in solving equations and applying these concepts in real-world scenarios.