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.

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> 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.       Log On


   

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) About Me  (Show Source):
You can put this solution on YOUR website!
Absolutely, let's break down this problem step-by-step.
**Given Information:**
1. logx(w) = 24
2. logx(yx) = 40
3. logxy²(zw) = 12
4. x, y, z > 1
5. w > 0
**Goal:** Find logz(w)
**Step 1: Simplify logx(yx)**
* logx(yx) = logx(y) + logx(x) = 40
* logx(y) + 1 = 40
* logx(y) = 39
**Step 2: Express w and y in terms of x**
* From logx(w) = 24, we get w = x24
* From logx(y) = 39, we get y = x39
**Step 3: Simplify logxy²(zw)**
* logxy²(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 logz(w)**
* logz(w) = logx924(x24)
* Using the property logab(cd) = (d/b)loga(c), we get:
* logz(w) = (24/924)logx(x)
* logz(w) = 24/924
* Simplify the fraction: 24 / 924 = 2 / 77
**Therefore, logz(w) = 2/77**