Problem 50 - Algebra 2

Question

Error Analysis Find the error in the following evaluation of log \(_{27} 3 .\) Then evaluate the logarithm correctly. $$ \begin{aligned} \log _{27} 3 &=x \\ 27 &=x^{3} \\ 3 &=x \\ \log _{27} 3 &=3 \end{aligned} $$

Step-by-Step Solution

Answer

The error occurs in the second step where \(27 = x^{3}\) is wrongly asserted. The correct value of the logarithm is \(x = 1/3\).

Step 1: Identifying the Error

Looking at the chronology, it starts correctly by expressing \(log_{27} 3\) as an equation \(27^x = 3\). The error occurs at the second step: \(27 = x^3\). Using the correct relationship, it should be \(27^x = 3\) not \(27 = x^3\). Hence, the third step, \(x = 3\) is also incorrect.

Step 2: Correct Calculation of the Logarithm

Now, we will correctly solve the logarithm \(log_{27} 3\). From the proper understanding of logarithm: if \(b^x = a\), then \(x = log_b a\), it follows that to find \(x\) in \(log_{27} 3 = x\) (which is the same as \(27^x = 3\)), we have to identify a number \(x\) such that \(27\) raised to the power \(x\) equals to \(3\). From the principles of exponents, we know that the cube root of \(27\) is \(3\). Equivalently, \(27^{1/3} = 3\). Hence, \(x = 1/3\).

Step 3: Checking the Answer

To ensure our answer is correct, we can substitute \(x=1/3\) into the equation \(27^x = 3\) to get \(27^{1/3}\). Evaluating this, indeed yields \(3\). So, our answer is correct.

Key Concepts

Error Analysis

Error analysis in math involves identifying mistakes in calculations or logic and figuring out how to correct them. It’s like being a detective, finding where things went wrong. In the exercise above, the error occurred because the relationship between base, exponent, and result was misunderstood. Instead of recognizing that we needed to find the power to which 27 must be raised to get 3, there was a misstep in rewriting this equation.
This type of mistake generally happens when rewriting the logarithmic equation into its exponential form. When we think about logarithms, we should remember that the logarithm of a number is the exponent to which the base needs to be raised to achieve that number. If errors occur during this conversion, it can lead to wrong results.
For this specific case:

• The original equation should have been: \(27^x = 3\).
• Instead, it was incorrectly expressed as \(27 = x^3\).
• By correcting this initial step, you set the foundation for the accurate solution.

By catching these errors, students can better understand the correct methods and improve their problem-solving skills.

Exponents

Exponents are a fundamental part of understanding logarithms. They show how many times you multiply a number by itself to get another number. For example, \(a^b\) means "multiply \(a\) by itself \(b\) times." Understanding this concept is crucial in solving the given problem.
Here are the key things to know about exponents:

• The base is the number being multiplied, and the exponent tells how many times to multiply the base. For instance, in \(27^x\), 27 is the base.
• Exponents can be whole numbers, fractions, or even negative numbers.
• If the exponent is a fraction, it indicates a root. So, \(27^{1/3}\) is finding the cube root of 27.
• Exponents are important in logarithmic functions because they define the result when the base is raised to this power.

In the case of \(log_{27} 3 = x\), knowing that \(27^{1/3} = 3\) informs us that the exponent is \(\frac{1}{3}\). Understanding these relationships clears up how we can move between exponential and logarithmic forms.

Logarithmic Equations

Logarithmic equations transition between multiplication and exponentiation, offering a inverse relationship to exponents. In simple terms, a logarithm finds the exponent or power of a given base that results in a specific number. When solving a problem like \(log_{27} 3\), we are finding the power \(x\) such that \(27^x = 3\).
This translation into a logarithmic framework involves:

• Identifying the base: Here, the base is 27.
• Understanding the result: In the problem, the outcome is 3.
• Expressing the equation: This results in the equation \(27^x = 3\), which changes to \(x = log_{27} 3\).

Logarithmic equations can simplify complex calculations, especially in growth and decay functions, or any situation where exponential growth occurs. Remember, solving logarithmic equations efficiently requires understanding both logarithms and exponents. This firm grip on their properties makes logical problem-solving easier and more intuitive.