Math Problem Statement
Solution
The equation presented is:
To find the sum of all solutions to this equation, we need to follow these steps carefully:
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Simplify the inner logarithmic expression: The term can be rewritten using the change of base formula:
Since , we get:
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Substitute into the equation: Now the equation becomes:
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Solve the outer logarithmic equation: To solve the equation, we need to deal with each logarithmic step and isolate .
I'll work through solving it now. sympy import symbols, Eq, log, solve
Define variables
x = symbols('x')
Define the equation
equation = Eq(log(log((7*x - 10) * (4 * log(2) / log(x)), 2), 2), 3)
Solve the equation
solutions = solve(equation, x) solutionsThe equation results in a complex expression involving the Lambert W function. It suggests that the solution may not have a simple closed form. However, the equation is solvable using advanced functions, and the solution is given as:
This expression involves the Lambert W function, which is commonly used to solve equations of this type. The exact numeric value can be approximated numerically.
Would you like further clarification on how to handle this expression or more details on the Lambert W function?
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Math Problem Analysis
Mathematical Concepts
Logarithms
Algebra
Change of Base Formula
Lambert W Function
Formulas
log_b(a) = log_c(a) / log_c(b)
log_b(x * y) = log_b(x) + log_b(y)
log_b(x^n) = n * log_b(x)
Theorems
Change of Base Theorem
Product Rule for Logarithms
Suitable Grade Level
Grades 11-12