To solve the equation [tex]\(\log (\cot 2) - \log (5) = 2 - 1\)[/tex], let's proceed step-by-step.
1. Simplify the right-hand side of the equation:
[tex]\[
2 - 1 = 1
\][/tex]
So, the equation now becomes:
[tex]\[
\log (\cot 2) - \log (5) = 1
\][/tex]
2. Combine the logarithmic terms on the left-hand side:
Using the property of logarithms that [tex]\(\log a - \log b = \log \left(\frac{a}{b}\right)\)[/tex], we can rewrite:
[tex]\[
\log (\cot 2) - \log (5) = \log \left(\frac{\cot 2}{5}\right)
\][/tex]
Therefore, the equation becomes:
[tex]\[
\log \left(\frac{\cot 2}{5}\right) = 1
\][/tex]
3. Exponentiate both sides to get rid of the logarithm:
Recall that if [tex]\(\log_b(x) = y\)[/tex], then [tex]\(x = b^y\)[/tex]. Assuming the logarithm is base 10 (common logarithm):
[tex]\[
\frac{\cot 2}{5} = 10^1
\][/tex]
[tex]\[
\frac{\cot 2}{5} = 10
\][/tex]
4. Solve for [tex]\(\cot 2\)[/tex]:
Multiply both sides of the equation by 5 to isolate [tex]\(\cot 2\)[/tex]:
[tex]\[
\cot 2 = 10 \times 5
\][/tex]
[tex]\[
\cot 2 = 50
\][/tex]
However, we need to verify if [tex]\(\cot 2 = 50\)[/tex] is valid.
5. Check the validity:
Knowing the behavior of the cotangent function and typical values, [tex]\(\cot(2)\)[/tex] is typically a small value (in the standard trigonometric range for angles measured in radians).
Given the above steps, we've found [tex]\(\cot 2 = 50\)[/tex]. But considering typical ranges of trigonometric functions, it seems like our equation has no solution when evaluated with realistic constraints and typical behaviors of trigonometric functions for the angles we use.
Thus, the conclusion based on the solution approach is that there is no valid solution within the typical range of parameter values for this problem.
