Can you Pass Harvard University Admission Interview ?

[@TomSkinner]

Why not take the log of both sides in the first place. Then you get x=log75/log45

[@dan-florinchereches4892]

I would say you obtain x ln(45)=ln(75) x( ln 9+ ln5)=ln3+ln25
x=(ln3+2ln5)/(2ln3+ln5) let y=ln5/ln3 then x= (1+2y)/(2+y)
substitute into required relation:
2+4y-2-y
-------------
2+y
(2x-1)/(x-2)= -------------- = 3y/(-3)=- ln5/ln3
1+2y-4-2y
---------
2+y
And I would stop there because i have aversion to calculators lol

[@lourdesgracia6767]

X = log 75 ÷ log 45

[@moseskimani7742]

The equation was too hard for him hence he run away😂😂

[@CharlesChen-el4ot]

45^x-1 = 5/3 = 10/6
(x-1)* log 45 = 1-log6
x -1 = (1-log6)/log 45
x = (1+log 45-log6)/log45
x = (1+log5+log3 -log2)/log45
= (2+log3-2log2)/
(1+log3-log2)
= 1 - log2/(1+log 3/2)

[@saschavogt5083]

I don’t understand. What‘s the value of x?

[@musicsubicandcebu1774]

It doesn't matter, that wasn't the question. Though you can find x first then sub into (2x-1)/(x-2).

[@andretewem3385]

Solutions in |C = ( 2.ln(5)+ln(3) ) / ( 2.ln(3)+ln(5) )
+ i.2.k.pi/( 2.ln(3)+ln(5) )
with k € Z.
With k = 0 we get solutions in |R : X = ln(75) / ln(45)

[@walterwen2975]

Harvard University Admission Interview: 45ˣ = 75, (2x - 1)/(x - 2) =?
45ˣ = 75, x > 0
First method:
Straight forward solution;
log(45ˣ) = log75, x = log₄₅75 = 1.1342
(2x - 1)/(x - 2) = [2(1.1342) - 1]/(1.1342 - 2) = 1.2684/(- 0.8658) = - 1.4650
Second method:
45ˣ = (9ˣ)(5ˣ) = (3²ˣ)(5ˣ) = 75 = (3)(25) = (3)(5²), (3²ˣ)/(3) = (5²)/(5ˣ)
3²ˣ⁻¹ = 5²⁻ˣ = (1/5)ˣ⁻², log(3²ˣ⁻¹) = log[(5⁻¹)ˣ⁻²], (2x - 1)log3 = (x - 2)log(5⁻¹)
(2x - 1)/(x - 2) = - log₃5 = - 1.465; 2x - 1 = - 1.465(x - 2), x = 1.134
Answer check:
45ˣ = 45¹·¹³⁴ = 75; Confirmed
The calculation was achieved on a smartphone with a standard calculator app
Final answer:
(2x - 1)/(x - 2) = - log₃5 = - 1.465

[@superacademy247]

Awesome 😎💯