Problem 4 - Elementary and Intermediate Algebra

Question

Fill in the blanks. \(\log _{x} 81=4\) is ____ to \(x^{4}=81\).

Step-by-Step Solution

Answer

Equivalent

Step 1: Identify Logarithmic Equation

We have the logarithmic equation given as \(\log_{x} 81 = 4\). This tells us that the logarithm base \(x\) of \(81\) equals \(4\). It asks us for what power we must raise \(x\) to get \(81\).

Step 2: Convert Logarithmic Equation to Exponential Form

To convert a logarithmic equation \(\log_{x} 81 = 4\) into an exponential form, we use the identity: If \(\log_{b}(a) = c\), then \(b^c = a\). Here, \(b = x\), \(a = 81\), and \(c = 4\). Thus, \(x^4 = 81\).

Step 3: Check the Equivalent Expression

On converting the equation, we confirmed that \(x^4 = 81\) is the exponential form of the given logarithmic equation. Hence, \(\log_{x} 81 = 4\) is equivalent to \(x^{4}=81\).

Key Concepts

Exponential Equations

Exponential equations are an essential part of mathematics, especially when dealing with growth, decay, and various real-world phenomena. Though they might seem complex, they follow a simple rule: you have a variable, typically represented as the base of a power, raised to an exponent that equals some constant. This format is written as \( b^c = a \), where:

• \(b\) is the base which is raised to a power
• \(c\) is the exponent–this tells us how many times to multiply the base \(b\) by itself
• \(a\) is the result or the value of the expression

In the context of exponential equations, the goal is often to find the base or the exponent. The key takeaway is that the exponent tells us the power to which the base must be raised to achieve the desired result. Understandably, as our example, for the equation \(x^4 = 81\), we find the value of \(x\) by solving the equation through various algebraic methods like taking roots.

Logarithmic Equations

Logarithmic equations are quite fascinating once understood. At their core, they are the inverse operations of exponential equations. This means if you have a logarithmic equation \(\log_{b}(a) = c\), it helps you find out the power (\(c\)) to which you must raise the base (\(b\)) in order to obtain \(a\).
Some key features of logarithmic equations are:

• The base in the logarithm, denoted by \(b\), must be a positive number
• The result or the output of the logarithm, \(c\), tells us about the power
• Just like exponential equations, logarithmic equations deal with the concept of inverses, making them fundamental in solving many algebraic problems

In the given example of \(\log_{x} 81 = 4\), you are essentially asking, "What power should \(x\) be raised to get \(81\)?" The result here makes it apparent that understanding logarithms allows you to solve for unknowns in a powerful way.

Conversion Between Forms

Converting between logarithmic and exponential forms is a crucial skill for solving equations in mathematics. This conversion is based fundamentally on the relationship that if \(\log_b(a) = c\), then \(b^c = a\).
Understanding this relationship allows you to switch between understanding and solving problems that involve exponential growth, decay, and many other real-world problems efficiently.
Here are some steps to watch out for when doing conversions:

• First, identify your \(b\), \(a\), and \(c\) from the logarithmic equation
• Apply the conversion identity \(\log_b(a) = c \rightarrow b^c = a\)
• Rewrite your equation using these identified elements

For instance, our example of converting \(\log_{x} 81 = 4\) to an exponential form follows this sequence quickly and easily into \(x^4 = 81\). This process not only facilitates solutions but also enhances your understanding of the relations between base, exponent, and result.