Answer :
Certainly! Let's go through the solution step by step and identify where the error occurs. Here is the given sequence of steps with the error:
[tex]\[ \begin{aligned} \log x - \log_5 3 & = 2 \log_5 3 \quad (\text{Step 1}) \\ \log x & = 3 \log_5 3 \quad (\text{Step 2}) \\ \log x & = \log_5 3^3 \quad (\text{Step 3}) \\ x & = 27 \quad (\text{Step 4}) \end{aligned} \][/tex]
Let's analyze each step for correctness:
1. Step 1: Simplification
[tex]\[ \log x - \log_5 3 = 2 \log_5 3 \][/tex]
The first step aims to isolate [tex]\(\log x\)[/tex]. However, note that on the left side, [tex]\(\log x\)[/tex] is not specific about the base. If we assume it's the natural logarithm ([tex]\(\log_e = \ln\)[/tex]) or the common logarithm ([tex]\(\log_{10}\)[/tex]), then there's an inconsistency in base comparison. Ensure that the logarithm bases are consistent.
2. Step 2: Rearranging and Combining Like Terms
[tex]\[ \log x = 3 \log_5 3 \][/tex]
To achieve this step correctly, the logarithm should be expressed with the same base. Since [tex]\(\log x\)[/tex] was moved and it's implicit, we should have:
[tex]\[ \log_5 x - \log_5 3 = 2 \log_5 3 \implies \log_5 x = 3 \log_5 3 \][/tex]
When we adjust it correctly, the steps align as:
[tex]\[ \log_5 x - \log_5 3 = 2 \log_5 3 \][/tex]
Isolating [tex]\(\log_5 x\)[/tex]:
[tex]\[ \log_5 x = 2 \log_5 3 + \log_5 3 \][/tex]
Simplifying further:
[tex]\[ \log_5 x = 3 \log_5 3 \][/tex]
3. Step 3: Logarithmic Property
[tex]\[ \log_5 x = \log_5 3^3 \][/tex]
Using the property of logarithms: [tex]\(\log_b (a^n) = n \log_b a\)[/tex], which confirms:
[tex]\[ \log_5 x = 3 \log_5 3 \implies \log_5 x = \log_5 27 \][/tex]
4. Step 4: Exponential Form
[tex]\[ x = 27 \][/tex]
When both sides have the same logarithmic expression with the same base, we can convert them into exponential form and directly conclude:
[tex]\[ x = 3^3 \implies x = 27 \][/tex]
Thus, the error in the initial provided steps was the inconsistency and ambiguity of the logarithmic base. By ensuring all logarithms are clearly represented with the same base, the solution correctly simplifies to:
[tex]\[ x = 27 \][/tex]
The additional step required was to clear up the base of the logarithms and use the exponentiation property correctly.
[tex]\[ \begin{aligned} \log x - \log_5 3 & = 2 \log_5 3 \quad (\text{Step 1}) \\ \log x & = 3 \log_5 3 \quad (\text{Step 2}) \\ \log x & = \log_5 3^3 \quad (\text{Step 3}) \\ x & = 27 \quad (\text{Step 4}) \end{aligned} \][/tex]
Let's analyze each step for correctness:
1. Step 1: Simplification
[tex]\[ \log x - \log_5 3 = 2 \log_5 3 \][/tex]
The first step aims to isolate [tex]\(\log x\)[/tex]. However, note that on the left side, [tex]\(\log x\)[/tex] is not specific about the base. If we assume it's the natural logarithm ([tex]\(\log_e = \ln\)[/tex]) or the common logarithm ([tex]\(\log_{10}\)[/tex]), then there's an inconsistency in base comparison. Ensure that the logarithm bases are consistent.
2. Step 2: Rearranging and Combining Like Terms
[tex]\[ \log x = 3 \log_5 3 \][/tex]
To achieve this step correctly, the logarithm should be expressed with the same base. Since [tex]\(\log x\)[/tex] was moved and it's implicit, we should have:
[tex]\[ \log_5 x - \log_5 3 = 2 \log_5 3 \implies \log_5 x = 3 \log_5 3 \][/tex]
When we adjust it correctly, the steps align as:
[tex]\[ \log_5 x - \log_5 3 = 2 \log_5 3 \][/tex]
Isolating [tex]\(\log_5 x\)[/tex]:
[tex]\[ \log_5 x = 2 \log_5 3 + \log_5 3 \][/tex]
Simplifying further:
[tex]\[ \log_5 x = 3 \log_5 3 \][/tex]
3. Step 3: Logarithmic Property
[tex]\[ \log_5 x = \log_5 3^3 \][/tex]
Using the property of logarithms: [tex]\(\log_b (a^n) = n \log_b a\)[/tex], which confirms:
[tex]\[ \log_5 x = 3 \log_5 3 \implies \log_5 x = \log_5 27 \][/tex]
4. Step 4: Exponential Form
[tex]\[ x = 27 \][/tex]
When both sides have the same logarithmic expression with the same base, we can convert them into exponential form and directly conclude:
[tex]\[ x = 3^3 \implies x = 27 \][/tex]
Thus, the error in the initial provided steps was the inconsistency and ambiguity of the logarithmic base. By ensuring all logarithms are clearly represented with the same base, the solution correctly simplifies to:
[tex]\[ x = 27 \][/tex]
The additional step required was to clear up the base of the logarithms and use the exponentiation property correctly.
