How to Solve Exponential Equations without Logarithms

[malco97]

I would be grateful if someone could tell me how to find unknown indices.

e.g 3^x=9

(i know it is 2 but i would like to know the process for use with larger numbers).

Thankyou in advance

 

[enslam]

3^x=9
therefore
9=3^2
get the log base 3 of both sides
x=2

 

[Lonewolf]

No, you used the fact x=2.

9 = 3^x

log 9 = log (3^x)

log 9 = x log 3

x = log9/log3 = 2

log can be to any base as long as they are the same. you will usually find base 10, or base e on your calculator. log base e is usually ln.

 

[KLscilevothma]

No, you used the fact x=2.

1) 3^x=9
As long as we can think of the relationship between 3 and 9, we can solve this problem without using logarithm, like the one suggested by enslam.

here are more examples (solve for x)
2) 5^x = 625
3) 4*5^x = 100
4) 2^x = 8

If you use Logarithm, in the step log9/log3, you need to know the fact that 9 = 3^2 too, unless you use a caculator.

 

[HallsofIvy]

1) 3^x=9
As long as we can think of the relationship between 3 and 9, we can solve this problem without using logarithm, like the one suggested by enslam.

That's true as long as the "opposite" problem, finding the power, can be done easily- as long as the answer is an integer as in all of your examples.

To solve, for example 3^x= 7, you will need to use logarithms.