Math Problem Statement

log2[log2(7x - 10) × logx 16] = 3, find sum of all solution to the equation

Solution

The equation presented is:

log2[log2((7x10)×logx16)]=3\log_2\left[ \log_2\left( (7x - 10) \times \log_x 16 \right) \right] = 3

To find the sum of all solutions to this equation, we need to follow these steps carefully:

  1. Simplify the inner logarithmic expression: The term logx16\log_x 16 can be rewritten using the change of base formula:

    logx16=log16logx\log_x 16 = \frac{\log 16}{\log x}

    Since log16=log(24)=4log2\log 16 = \log (2^4) = 4 \log 2, we get:

    logx16=4log2logx\log_x 16 = \frac{4 \log 2}{\log x}

  2. Substitute into the equation: Now the equation becomes:

    log2[log2((7x10)×4log2logx)]=3\log_2 \left[ \log_2 \left( (7x - 10) \times \frac{4 \log 2}{\log x} \right) \right] = 3

  3. Solve the outer logarithmic equation: To solve the equation, we need to deal with each logarithmic step and isolate xx.

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:

x=64W(log(2684354562(27/32)/512))7log(2)x = -\frac{64 \cdot W\left( -\log(268435456^{2^{(27/32)}/512}) \right)}{7 \cdot \log(2)}

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