Problem 17 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the exponential equation in logarithmic form. For example, the logarithmic form of \(2^{3}=8\) is \(\log _{2} 8=3.\) $$81^{1 / 4}=3$$

Step-by-Step Solution

Answer

The logarithmic form of the given exponential equation \( 81^{1 / 4} = 3 \) is \( \log _{81} 3 = 1/4 \).

Step 1: Identify Elements

First of all, identify the base, the exponent, and the result of the exponentiation. Here, in the equation \( 81^{1 / 4}=3 \), the base is 81, the exponent (or power) is 1/4, and the result of the exponentiation (or the number the base is raised to) is 3.

Step 2: Convert into Logarithmic Form

Next, we transform the given exponential equation into its equivalent logarithmic form. According to the principle \( \log _{b} a = c \), where 'b' is the base, 'a' is the result, and 'c' is the exponent, our equation becomes \( \log _{81} 3 = 1/4 \).

Key Concepts

Logarithmic Form

Logarithmic form is a way of expressing exponential equations in a different manner. When we talk about logarithms, we're essentially asking, "To what power must we raise the base to get a certain number?" This is the inverse operation of exponentiation. As with any mathematical language, once you learn the basic rules, you can translate it back and forth. For example, in the equation \(81^{1/4} = 3\), we have an exponential equation. To convert it to logarithmic form, we apply the principle: \(\log_{b} a = c\). Here:

• \(b = 81\) is the base of the exponent.
• \(a = 3\) is the result or the outcome of the exponentiation.
• \(c = \frac{1}{4}\) is the exponent itself, indicating the power to which the base must be raised.

So, the logarithmic form of the exponential statement \(81^{1/4} = 3\) turns into \(\log_{81} 3 = \frac{1}{4}\). Understanding this form helps evaluate and solve more complex equations.

Base and Exponent

In any exponential equation, the 'base' is a critical component, representing the number that is being raised to a power. An exponential equation follows the pattern of \(b^c = a\), where:

• \(b\) is the base that we repeatedly multiply.
• \(c\) is the exponent, also known as the power or index, indicating the number of times the base is used.
• \(a\) is the result of these operations, sometimes called the power of the number.

For our example, \(81^{1/4} = 3\), \(81\) acts as the base, the number 1/4 is the exponent. This exponent tells us that we are seeking a quantity which when the base is raised to it, gives us 3. This concept is fundamental for understanding how exponents work within equations. It ensures that we can manipulate numbers in various mathematical contexts.

Conversion of Equations

The conversion of an exponential equation to a logarithmic form (and vice versa) is a critical mathematical skill. It involves changing the way an equation is represented but keeping the information intact. The notion is simple: in an exponential form, equations are expressed as \(b^c = a\). In logarithmic form, this same equation translates to \(\log_{b} a = c\).

Let's see how this applies to our exercise:
When you see \(81^{1/4} = 3\), you're looking at an exponential expression. By converting it, you express it as \(\log_{81} 3 = \frac{1}{4}\).

• The original base of the exponent \(81\) becomes the base of the logarithm.
• The result \(3\) of the exponential expression turns into the number that the logarithm is taken of.
• Finally, the exponent \(\frac{1}{4}\) is the answer to the logarithmic equation.

This conversion process is much like translating between two languages. It enables us to solve problems in different ways, providing flexibility in mathematical problem-solving.