SOLUTION: Let x, y, and z all exceed 1, and let w be a positive number such that \log_x w = 24, \log_x yx = 40, and \log_{xy^2} zw = 12. Find \log

SOLUTION: Let x, y, and z all exceed 1, and let w be a positive number such that \log_x w = 24, \log_x yx = 40, and \log_{xy^2} zw = 12. Find \log_z w.

Question 1209847: Let x, y, and z all exceed 1, and let w be a positive number such that \log_x w = 24, \log_x yx = 40, and \log_{xy^2} zw = 12. Find \log_z w.

Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
Absolutely, let's break down this problem step-by-step.
**Given Information:**
1. log_x(w) = 24
2. log_x(yx) = 40
3. log_xy²(zw) = 12
4. x, y, z > 1
5. w > 0
**Goal:** Find log_z(w)
**Step 1: Simplify log_x(yx)**
* log_x(yx) = log_x(y) + log_x(x) = 40
* log_x(y) + 1 = 40
* log_x(y) = 39
**Step 2: Express w and y in terms of x**
* From log_x(w) = 24, we get w = x²⁴
* From log_x(y) = 39, we get y = x³⁹
**Step 3: Simplify log_xy²(zw)**
* log_xy²(zw) = 12
* zw = (xy²)¹²
* zw = x¹²(y²)¹²
* zw = x¹²y²⁴
**Step 4: Substitute y and w in terms of x**
* z(x²⁴) = x¹²(x³⁹)²⁴
* z(x²⁴) = x¹²x⁹³⁶
* z(x²⁴) = x⁹⁴⁸
* z = x⁹⁴⁸ / x²⁴
* z = x⁹²⁴
**Step 5: Find log_z(w)**
* log_z(w) = log_x⁹²⁴(x²⁴)
* Using the property log_a^b(c^d) = (d/b)log_a(c), we get:
* log_z(w) = (24/924)log_x(x)
* log_z(w) = 24/924
* Simplify the fraction: 24 / 924 = 2 / 77
**Therefore, log_z(w) = 2/77**

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