Problem 9 - College Algebra with Corequisite Support

Question

For the following exercises, rewrite each equation in exponential form. \(\log _{x}(64)=y\)

Step-by-Step Solution

Answer

The exponential form is \( x^y = 64 \).

Step 1: Identify the Logarithmic Form

The given equation is \( \log_{x}(64) = y \). This is in the form \( \log_{b}(a) = c \), where \( b \) is the base, \( a \) is the argument or result inside the logarithm, and \( c \) is the exponent.

Step 2: Convert to Exponential Form

Using the property of logarithms that relates to exponents, convert the equation \( \log_{x}(64) = y \) into the exponential form. This property states that if \( \log_{b}(a) = c \), then it can be rewritten as \( b^c = a \).

Step 3: Write the Exponential Equation

Following Step 2, rewrite the logarithmic equation \( \log_{x}(64) = y \) in its equivalent exponential form. Therefore, it becomes \( x^y = 64 \). This equation states that \( x \) raised to the power \( y \) equals 64.

Key Concepts

Logarithmic Form

Understanding how logarithms work is essential in mathematics. The logarithmic form is a way of expressing an equation where you are trying to find the power to which a certain number, called the base, must be raised to obtain another number.
For example, in the logarithmic expression \( \log_{b}(a) = c \), \( b \) represents the base, \( a \) stands for the result, and \( c \) is the logarithm or the exponent required to obtain \( a \) from \( b \). It rearranges the concept of exponentiation into this unique format that is inversely related to exponential form.
Logarithms are quite useful, especially when dealing with very large numbers, as they allow you to simplify complex multiplication and division into addition and subtraction, respectively.

• Base: This is the number you start with and perform repeated multiplication on.
• Argument: This is the result or the number you want to find out how many times the base must be multiplied to yield it.
• Logarithm: This represents the exponent needed to achieve the argument from the base.

Exponential Equation

An exponential equation involves numbers and variables being raised to a power. Essentially, it deals with expressions that contain exponents. In such equations, the base is raised to the power of the exponent yielding a particular result.
An important property of exponential equations is that they can be transformed from and into logarithmic forms, which can simplify solving them when the variable is an exponent.
For example, converting \( \log_{x}(64) = y \) into exponential form gives us \( x^y = 64 \). Here, \( x \) is the base raised to the power \( y \) to produce 64.
This form is particularly useful when you are given a logarithmic equation and need to solve for a base or exponent, as converting it to an exponential equation can sometimes make it easier to understand or solve.

• Base: The constant or number that is raised to an exponent.
• Exponential form: The expression of the relationship using exponents.
• Result: The outcome of raising the base to the given power.

Properties of Logarithms

Logarithms possess unique properties that make them powerful tools in mathematics. They adhere to specific rules that allow for easier manipulation of equations involving exponents and can simplify complex algebraic expressions.
Some key properties of logarithms include:

• Product Rule: \( \log_{b}(mn) = \log_{b}(m) + \log_{b}(n) \). This means the logarithm of a product is the sum of the logarithms.
• Quotient Rule: \( \log_{b}\left(\frac{m}{n}\right) = \log_{b}(m) - \log_{b}(n) \). This indicates that the logarithm of a quotient is the difference between the logarithms.
• Power Rule: \( \log_{b}(m^n) = n \cdot \log_{b}(m) \). This property shows that the logarithm of a power is the exponent times the logarithm of the base number.

These properties are valuable for simplifying and solving equations involving logarithmic terms, especially in cases where multiple logarithms can be combined or broken down, easing the solving process.