MHB Solve Exponent Problem: 3^x=4^y=12^z

  • Thread starter Thread starter Bushy
  • Start date Start date
  • Tags Tags
    Exponent
Click For Summary
To solve the equation 3^x = 4^y = 12^z, it is essential to express the relationships between the bases. By taking logarithms, one can derive that 3^(xy/(x+y)) equals 12^(zy/(x+y)). Recognizing that both 3 and 4 are factors of 12 helps simplify the problem, leading to the conclusion that z can be expressed as z = (xy)/(x+y). This relationship highlights the interconnectedness of the variables x, y, and z in the context of exponentiation. The discussion emphasizes the importance of using properties of logarithms and factor relationships to solve complex exponent problems.
Bushy
Messages
40
Reaction score
0
Given [Math]3^x=4^y =12^z[/Math] show that $$z=\frac{xy}{x+y}$$

I've take logs on both sides and find myself going in circles, any hints?
 
Mathematics news on Phys.org
$$3^{xy/(x+y)}=12^{zy/(x+y)}$$

Now try using the fact that $3$ and $4$ are factors of $12$ and that $4^y=3^x$.
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
View full post »

Similar threads

MHB Show x³y+y³z+z³x is a constant
  • · Replies 1 ·
Replies
1
Views
1K
MHB Solving System of Equalities: x^2-y\sqrt xy & y^2-x\sqrt xy =3
  • · Replies 2 ·
Replies
2
Views
1K
MHB Solve System of 4 Equalities: x+y+z+w=22, xyzw=648
  • · Replies 6 ·
Replies
6
Views
1K
MHB What is the Sum of x, y, and z in a Non-Negative Real Number System?
  • · Replies 4 ·
Replies
4
Views
2K
MHB Prove Divisibility: $(x-y)^2+(y-z)^2+(z-x)^2=xyz$ yields $x^3+y^3+z^3$
  • · Replies 1 ·
Replies
1
Views
2K
MHB Solve System of Equalities: x^2+xy+y^2, x^2+xz+z^2, y^2+yz+z^2
  • · Replies 7 ·
Replies
7
Views
2K
MHB Is There an Exact Solution to the Equation \(4^x + 3^x = 12\)?
  • · Replies 7 ·
Replies
7
Views
1K
MHB Solving $x^3+y^3=7$ and $x^2+y^2+x+y+xy=4$ System of Equations
  • · Replies 1 ·
Replies
1
Views
1K
MHB Factorize 6(x^5+y^5+z^5)-5(x^2+y^2+z^2)(x^3+y^3+z^3)
  • · Replies 1 ·
Replies
1
Views
1K
MHB Challenge Problem #7: Σ(x/(y^3+2))≥1
  • · Replies 2 ·
Replies
2
Views
2K