Solving Exponential Growth Equation without Logarithms

[thomasrules]

the question is how do I solve

$$(1.024)^t=2.857$$

and find "t" without using logarithms

 

[Office_Shredder]

You don't.

You could just guess and check... i.e., what is it if t=2, 3, 4, etc. If at t₀ the LHS is under 2.857, and at t₁ the LHS is over, you know that the t you're looking for is between the two values.

There's probably an algorithm you can use or something

 

[HallsofIvy]

Why would you not want to use logarithms? That's what logarithms are for! Other than that, use the "midpoint algorithm". Find two values of t that give one value lower than 2.857 and one larger (hint: try 44 and 45), then try half way between. Keep going "half way" between one number that gives less than 2.857 and one that gives more than, reducing the interval each time.

 

[Checkfate]

Gotta just guess and guess and guess. It's 44.2633... lol :).

But... Just so you know in the future. $$A^{x}=A^{y}$$ can be rewritten as x=y.

Edit, in this case $$(1.024)^{t}=(1.024)^{44.2633}$$ So $$t=44.2633$$

 

[thomasrules]

yea i know that rule thanks guys

 

[thomasrules]

yea ok but wait how do u find the inverse of like

$$y=3(2)^x$$ or $$y=(x)^{1/3}$$

whats the formula is not in the book

 

[thomasrules]

i suck at this tex stuff i can't get the x^(1/3) and I wrote the y=3(2)^x first and it appeared second...wtf

 

[mk_gm1]

Do you mean inverse? The inverse of a function basically just means swap x and y around. So $$y=x^{1/3}$$ goes to $$x=y^{1/3}$$...