Problem 6 - College Algebra

Question

Write each equation in its equivalent exponential form. $$ 3=\log _{b} 27 $$

Step-by-Step Solution

Answer

The exponential form of the equation, \(3 = \log _{b}(27)\), is \(b^{3} = 27 \)

Step 1: Understand the components

The base of the logarithm is 'b', the logarithm itself is '3', and '27' is the number we take the logarithm of.

Step 2: Apply the principle of conversion from logarithmic to exponential form

The principle \(b^{log_b{Y}} = Y\) is used to convert the logarithmic equation into exponential form. 'b' is the base, '3' is the exponent, and '27' is the result. Therefore, \(b^{3} = 27 \).

Key Concepts

Logarithm Basics

Logarithms are like a different way to express exponentiation, allowing us to work with very large or very small numbers in a more manageable form. When we say that \( 3=\log_b 27 \), we are saying that the base \( b \) raised to the power of 3 equals 27.

The general form of logarithm is \( \log_b X = Y \), which reads as: 'log base b of X equals Y'. The key components here are the base \( b \), the result \( Y \), and the argument \( X \). In the exercise, \( b \) is the base we are looking for, 3 is our result \( Y \), which is the power the base is raised to, and 27 is our argument \( X \).

Exponential Equations

An exponential equation is one in which variables appear as exponents, like in \( b^3 = 27 \) from our example. These equations can often be solved by rewriting them in logarithmic form and vice versa. The ability to convert between logarithmic and exponential forms is a foundational skill in algebra. Understanding this relationship is crucial, as it allows us to solve equations involving exponentiation and logarithms more easily.

For instance, if we are given the exponential equation \( b^3 = 27 \), we can use the laws of exponents to find that the base \( b \) must be the cube root of 27, giving us the solution \( b = 3 \). This is because raising 3 to the third power equals 27.

Algebraic Expressions

An algebraic expression is a collection of numbers, variables, and operations. Transforming a logarithmic equation into an exponential form is an example of working with algebraic expressions. It can also involve simplifying, expanding, factoring, or otherwise manipulating expressions to solve equations or understand their properties.

In the given exercise, we manipulate the logarithmic expression to derive its equivalent exponential form. This involves recognizing the components of the logarithmic expression and applying algebraic rules to restate the equation. We've simplified the expression \( 3=\log_b 27 \) to the algebraic expression \( b^3 = 27 \), revealing the relationship between these elements in a different, yet equivalent, form.