## anonymous one year ago I've been having a touch of trouble with solving logarithms with uneven bases. For example 8^x-10=3^x+2 Now I know that in order to solve I need to log8^x-10=log3^x+2 then you move the power out in front of the logs like so x-10 log8=x+2 log3 this is where I stumble, you find the approximate value of the log soo (x-10)(.903)=(x-2)(4.77) It's after this step were I get lost.

1. phi

if it was (x-10)*2 = (x-2)*3 what would you do? I would distribute, and combine "like terms"

2. phi

btw, I assume you meant log(3) = 0.47712 (not 4.77)

3. phi

so do this $(x-10)(.903)=(x-2)(0.477) \\ 0.903x -9.03 = 0.477x -0.954$

4. phi

can you finish?

5. anonymous

Yea one sec

6. anonymous

.426x=-.051 -/051/.426=-.119

7. anonymous

Except x is supposed to = 23.44 ... So I did something wrong

8. phi

ok, then step by step $8^{(x-10)}=3^{(x+2)} \\ \log\left( 8^{(x-10)}\right) = \log\left( 3^{(x+2)} \right)$ $(x-10) \log(8) = (x+2) \log(3)$ rather than approximate the logs, treat them as numbers (which is what they are) $x \log(8) - 10\log(8) = x \log(3) + 2 \log(3) \\ x \log(8) - x \log(3) = 10\log(8) + 2 \log(3) \\ x ( \log(8) - \log(3) ) = 10\log(8) + 2 \log(3) \\$ and finally $x = \frac{10\log(8) + 2 \log(3) }{ \log(8) - \log(3)}$

9. phi

btw, my first equation I managed to write (x-2) instead of x+2. That was bad. but, do you follow the steps up above? to get the final answer you need a calculator

10. anonymous

I get 10.57.... not 23.44

11. phi

what do you get for the top? (btw, I'm getting 23.44102, so it does work out)

12. anonymous

I type in 10log(8)+2log(3) and get 9.985...

13. phi

ok, now the bottom

14. anonymous

.4259...

15. phi

I get 0.425969 which rounds to 0.4260 to 4 digits but ok now 9.985/ 0.426

16. phi

if you keep more digits , you will get a more accurate answer

17. phi

if you type the expression into google like this (10 log 8 + 2 log 3)/(log 8 - log 3)= you get out 23.441021894

18. phi

but I would learn how to use a calculator (it is a useful to know how to do complicated expressions)