MHB Exponential Equation solve 54⋅2^(2x)=72^x⋅√0.5

  • Thread starter Thread starter Yankel
  • Start date Start date
  • Tags Tags
    Exponential
AI Thread Summary
The exponential equation 54⋅2^(2x)=72^x⋅√0.5 is analyzed, with the goal of solving for x without using logarithms. The discussion highlights the need to equate the exponents of both sides, leading to the equations 2x + 1 = x - 1/2 and 3 = 2x. Initial attempts reveal inconsistencies in the solutions derived from these equations. However, a corrected approach simplifies the equation to show that 18^3 = 18^(2x), resulting in the solution x = 3/2. The final answer is confirmed as 3/2, resolving the problem.
Yankel
Messages
390
Reaction score
0
Hello all,

I need assistance in solving this exponential equation.

\[54\cdot 2^{2x}=72^{x}\cdot \sqrt{0.5}\]

The final answer should be 3/2.

My strategy was to try and bring to a state where the exponents are equal. We know that 54 is 6 times 9. We also know 72 is 8 times 9. The solution probably involves the fact that 9 appears in both numbers.

Can you kindly assist ? Oh, one more thing, important thing. The use of logarithms is forbidden... :-)

Thank you .
 
Mathematics news on Phys.org
Yankel said:
Hello all,

I need assistance in solving this exponential equation.

\[54\cdot 2^{2x}=72^{x}\cdot \sqrt{0.5}\]

The final answer should be 3/2.

My strategy was to try and bring to a state where the exponents are equal. We know that 54 is 6 times 9.
More to the point, 54 is 2 times 27: 54= 2(3^3)

We also know 72 is 8 times 9.
Yes, and that is 72= (2^3)(3^2). Of course \sqrt{0.5}= \frac{1}{\sqrt{2}}= 2^{-1/2}

The solution probably involves the fact that 9 appears in both numbers.

Can you kindly assist ? Oh, one more thing, important thing. The use of logarithms is forbidden... :-)

Thank you .
54(2^{2x})= 2(3^3)(2^{2x})= 2^{2x+1}(3^3) and 72^x\sqrt{0.5}= 2^x(3^{2x})2^{-1/2}= 2^{x- 1/2}3^{2x}

54(2^{2x})=72^x\sqrt{0.5} is the same as
2^{2x+1}(3^3)= 2^{x- 1/2}3^{2x}

But now we have a problem! In order for those to be equal the exponents of both 2 and 3 must be the same on each side. We must ave both 2x+ 1= x- 1/2 and 3= 2x. To solve 2x+ 1= x- 1/2, subtract x and 1 from both sides: x= -3/2. To solve 3= 2x divide both sides by 2: x= 3/2. Those are NOT the same! There is no value of x that satisfies this.
 
Thank you very much.

I think that you have a small mistake with the exponents at the beginning but the general approach helped me get to the correct solution.
 
$$54\cdot 2^{2x}=72^{x}\cdot \sqrt{0.5}$$

$$54\cdot 4^x=18^{x}\cdot4^x\cdot\sqrt{\frac12}$$

$$54=18^x\cdot\sqrt{\frac12}$$

$$2\cdot54^2=18^{2x}$$

$$2\cdot3^2\cdot18^2=18^{2x}$$

$$18^3=18^{2x}\implies x=\frac32$$
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...

Similar threads

Replies
1
Views
1K
Replies
6
Views
1K
Replies
7
Views
1K
Replies
2
Views
1K
Replies
2
Views
2K
Replies
3
Views
1K
Replies
2
Views
1K
Back
Top