Verified Answer :

  • The algebraic expansion of \(126_n = 86_n\) results in \(n^2 - 6n = 0\), which gives \(n = 0\) or \(n = 6\).
  • Because the digit \(8\) is used in the number \(86_n\), the base \(n\) must be greater than \(8\), meaning no valid integer solution exists.
  • The exponential equation \(5^3 = 125\) is equivalent to the logarithmic form \(\log_5(125) = 3\).
  • Final results: No solution for \(n\); logarithmic form \(\boxed{\log_5(125) = 3}\).

Explanation

  1. Solving the Base Equation

For Part 1, we are asked to find a base \(n\) such that the equality \(126_n = 86_n\) holds. In base \(n\), a number \(d_k d_{k-1} \dots d_0\) represents the value \(\sum_{i=0}^k d_i n^i\).
Applying this to our given numbers:

  • \(126_n = 1 \cdot n^2 + 2 \cdot n^1 + 6 \cdot n^0 = n^2 + 2n + 6\)
  • \(86_n = 8 \cdot n^1 + 6 \cdot n^0 = 8n + 6\)

Setting these expressions equal, we have:
\[n^2 + 2n + 6 = 8n + 6\]
Subtracting \(8n + 6\) from both sides gives:
\[n^2 - 6n = 0\]
Factoring the equation:
\[n(n - 6) = 0\]
This yields solutions \(n = 0\) or \(n = 6\). However, in any base \(n\), all digits must be strictly less than the base \(n\). Here, the number \(86_n\) contains the digit \(8\), which necessitates that \(n > 8\). Since neither \(0\) nor \(6\) satisfies \(n > 8\), there is no base \(n\) for which \(126_n = 86_n\).

  1. Converting Exponential to Logarithmic Form

For Part 2, we are given the exponential equation \(5^3 = 125\). We know that the definition of a logarithm states that for \(b^e = v\), the equivalent logarithmic form is \(\log_b(v) = e\).
Comparing \(5^3 = 125\) to the general form:

  • The base \(b\) is \(5\).
  • The exponent \(e\) is \(3\).
  • The value \(v\) is \(125\).

Substituting these values into the logarithmic form, we obtain:
\[\log_5(125) = 3\]

  1. Conclusion

In summary, the equation \(126_n = 86_n\) has no valid base solution \(n\) because the requirement that the base must be greater than its largest digit (specifically \(n > 8\)) is not met by the algebraic solutions. The logarithmic form of the equation \(5^3 = 125\) is \(\log_5(125) = 3\). The final results are that there is no solution for \(n\) and the logarithmic form is \(\log_5(125) = 3\).

Examples

Understanding bases is crucial in computer science, where data is represented in binary (base 2), octal (base 8), or hexadecimal (base 16). For example, a color on a webpage might be represented in hexadecimal as #FFFFFF. Similarly, logarithms are used in real-life measurements like the Richter scale for earthquakes or the pH scale for acidity, allowing us to represent very large or very small numbers in a manageable way. Mastery of these conversions helps in analyzing data and simplifying complex equations in various scientific fields.

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