Problem 16 - College Algebra with Corequisite Support

Question

For the following exercises, rewrite each equation in logarithmic form. \(4^{x}=y\)

Step-by-Step Solution

Answer

\(\log_4 y = x\)

Step 1: Identify the Base, Exponent, and Result

In the given equation, identify the base of the exponent, the exponent itself, and the result. Here, the base is 4, the exponent is \(x\), and the result is \(y\).

Step 2: Apply the Definition of Logarithms

Recall that if \(a^b = c\), then it can be rewritten in logarithmic form as \(\log_a c = b\). Apply this definition to the equation \(4^x = y\).

Step 3: Write the Logarithmic Form

Using the definition from Step 2, rewrite \(4^x = y\) as \(\log_4 y = x\). This expression means that \(x\) is the exponent you raise 4 to in order to get \(y\).

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. The base is the number that gets multiplied, and the exponent tells how many times to multiply the base by itself. The expression is written as \(a^b\), where \(a\) is the base and \(b\) is the exponent.
Here are some important points to remember about exponentiation:

• A positive exponent indicates multiplication: \(a^3 = a \cdot a \cdot a\)
• Any number raised to the power of 0 equals 1: \(a^0 = 1\) (as long as \(a eq 0\))
• Exponents influence the direction and size of growth, where larger exponents result in larger numbers

Exponentiation plays a key role in understanding logarithms, which are essentially the inverse operation. In the context of \(4^x = y\), the operation indicated that the base \(4\) is raised to the power of \(x\) to yield the result \(y\). Understanding this process is crucial when shifting into logarithmic transformations.

Logarithms

Logarithms are the mathematical operation that undoes exponentiation. If you have the equation \(a^b = c\), logarithms help find what \(b\) must be to yield \(c\) when raising \(a\) to that power. The logarithmic form is written as \(\log_a c = b\).
Here are some basic properties of logarithms:

• \(\log_a a = 1\), because if the base and the number are the same, the exponent is 1.
• \(\log_a 1 = 0\), since any number raised to the power 0 equals 1.
• The logarithm of a product can be expressed as the sum of logarithms: \(\log_a (xy) = \log_a x + \log_a y\).

In the example equation \(4^x = y\), using logarithms, we transform it into \(\log_4 y = x\). Thus, \(x\) is determined as the exponent to which the base \(4\) must be raised to result in \(y\). Understanding the link between exponentiation and logarithms aids in solving many mathematical problems.

Equation Transformation

Equation transformation involves changing the form of an equation without altering its inherent meaning. This process is crucial in solving and understanding equations by moving between exponentials and logarithms depending on the context.
Here's what transformation entails:

• Identifying the elements of the original equation: base, exponent, and result.
• Reapplying known mathematical properties to restructure the equation, keeping the original values equal.
• Using definitions: moving from an exponent form \(a^b = c\) to a logarithm form \(\log_a c = b\).

For example, transforming \(4^x = y\) to \(\log_4 y = x\) involves understanding both the structure of exponentials and logarithms. This transformation provides a method to solve for variables in one form using another form, thereby enhancing analytical techniques in algebra. Transformations are powerful tools for dissecting complex equations and finding solutions efficiently.