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Answer :
To solve the equation [tex]\(\log _{1.3} 2 \cdot 197 = x\)[/tex], we need to find the value of [tex]\(x\)[/tex] by following these steps:
1. Understand the Explanation of Logarithms:
- The expression [tex]\(\log_{1.3} 2\)[/tex] is asking for the power to which we must raise the base [tex]\(1.3\)[/tex] to get [tex]\(2\)[/tex].
2. Calculate the Logarithmic Value:
- To find [tex]\(\log_{1.3} 2\)[/tex], we interpret it as [tex]\(\frac{\log_{10} 2}{\log_{10} 1.3}\)[/tex] using the change of base formula for logarithms, where [tex]\(\log_{10}\)[/tex] is the logarithm with base 10.
3. Multiply the Logarithmic Value by 197:
- Once we have the value of [tex]\(\log_{1.3} 2\)[/tex], we multiply it by 197 to find [tex]\(x\)[/tex].
4. Final Result:
- After calculating, we find that [tex]\(\log_{1.3} 2 \approx 2.6419\)[/tex].
- When [tex]\(2.6419\)[/tex] is multiplied by 197, the resulting value of [tex]\(x\)[/tex] is approximately 520.46.
Therefore, the value of [tex]\(x\)[/tex] is approximately 520.46.
1. Understand the Explanation of Logarithms:
- The expression [tex]\(\log_{1.3} 2\)[/tex] is asking for the power to which we must raise the base [tex]\(1.3\)[/tex] to get [tex]\(2\)[/tex].
2. Calculate the Logarithmic Value:
- To find [tex]\(\log_{1.3} 2\)[/tex], we interpret it as [tex]\(\frac{\log_{10} 2}{\log_{10} 1.3}\)[/tex] using the change of base formula for logarithms, where [tex]\(\log_{10}\)[/tex] is the logarithm with base 10.
3. Multiply the Logarithmic Value by 197:
- Once we have the value of [tex]\(\log_{1.3} 2\)[/tex], we multiply it by 197 to find [tex]\(x\)[/tex].
4. Final Result:
- After calculating, we find that [tex]\(\log_{1.3} 2 \approx 2.6419\)[/tex].
- When [tex]\(2.6419\)[/tex] is multiplied by 197, the resulting value of [tex]\(x\)[/tex] is approximately 520.46.
Therefore, the value of [tex]\(x\)[/tex] is approximately 520.46.
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