Problem 130 - Precalculus

Question

For the following exercises, rewrite each equation in exponential form. $$\log _{16}(y)=x$$

Step-by-Step Solution

Answer

The exponential form is \(16^x = y\).

Step 1: Identify the Logarithmic Components

In the equation \(\log_{16}(y) = x\), we identify the base of the logarithm, which is 16, the argument of the logarithm, which is \(y\), and the result of the logarithm, which is \(x\).

Step 2: Convert to Exponential Form

To rewrite a logarithmic equation in exponential form, we use the definition of a logarithm: if \(\log_b(a) = c\), it can be rewritten as \(b^c = a\). Applying this to our equation \(\log_{16}(y) = x\), we convert it to exponential form as \(16^x = y\).

Key Concepts

Logarithmic Equation

A logarithmic equation is a mathematical expression that involves logarithms. In these equations, you are typically solving for an unknown variable within the context of the logarithmic function. For instance, in the equation \( \log_{16}(y) = x \), the logarithm Operator is \( \log_{16} \), and it operates on \( y \), producing the result \( x \).
This type of equation is useful for solving problems where the variable of interest is inside a logarithmic function and can be complex because it often requires understanding both the base of the logarithm and its inverse operation, exponentiation.

Base of the Logarithm

The base of a logarithm is a critical component in understanding and solving logarithmic equations. It is the number that is raised to a power to obtain a given number. In our example, the equation \( \log_{16}(y) = x \) has a base of 16.

• This means each time you apply the logarithm, you are asking the question: "To what power must 16 be raised to yield \( y \)?"
• The base is always positive and not equal to 1, which avoids undefined or trivial results.

In practical problems, choosing the right base can simplify the understanding of the equation. The base helps determine how the logarithm will scale the results.

Exponential Equation

An exponential equation is one where variables appear as exponents. This is the opposite or inverse form of a logarithmic equation. When you write \( b^c = a \), it indicates that the base \(b\) raised to the power of \(c\) equals \(a\).
Using our previous logarithmic equation \( \log_{16}(y) = x \), converting it to its exponential form results in \( 16^x = y \). Here:

• 16 is the base,
• \(x\) is the exponent,
• \(y\) is the result.

Recognizing and working with exponential equations is crucial because they frequently model real-world phenomena, such as population growth, radioactive decay, and interest calculations.

Convert to Exponential Form

Converting a logarithmic equation to exponential form involves using the definition of logarithms. If you have \( \log_b(a) = c \), it can simply be rewritten as \( b^c = a \). In our example, \( \log_{16}(y) = x \) is directly transformed to \( 16^x = y \).
This conversion process can demystify logarithmic equations:

• It helps translate the logarithmic terms into a more familiar form of exponential equations.
• It allows for easier manipulation of equations involving exponents.
• This is especially beneficial in calculus and algebra, where such conversions often simplify the solving process.

By mastering this conversion, you can tackle a wide range of problems involving logarithms and exponentials more effectively.