How do you solve logarithms?

Question

Here are some examples of questions I have problems with:
1. log(x+2) + log(x-2) = log7

2. 2log(base2)^x-log(base2)(x-1) = 3

3. log(x-3)+logX= log7

Answer 1

(1) (x+2)(x-2) = 7

(2) 2^3/2 = x/(x-1)

(3) x(x-2) = 7

Answer 2

Use the properties of logs first to get one log OR one on each side. If it's one log = a number, then use the definition. If it's one on each side, then their parts are equal.

#1) log(x+2) + log (x-2) = log 7
log [(x+2)(x-2)] = log 7
x^2 - 4 = 7
x^2 = 11
x = sqrt(11)

#2) 2log(base2)x - log(base2)(x-1) = 3
log [x^2/(x-1)] = 3
2^3 = x^2/(x-1)
8 = x^2/(x-1)
8x - 8 = x^2
(can you take it from here?)

This is like #1:
log [x(x-3)] = log 7
x(x-3) = 7
and you can take it from here, I'm sure!

that's it! :)

Answer 3

Question 1
log [(x + 2) (x - 2) ] = log 7
(x + 2) (x - 2) = 7
x² - 4 = 7
x² = 11
x = √11

Question 2
log is log to base 2 :-
2 log x - log (x - 1) = 3
log x² - log (x - 1) = 3
log [ x² / (x - 1) ] = 3
[ x ² / (x - 1) ] = 2 ³
x ² = 8 (x - 1)
x ² - 8x + 8 = 0
x = [8 ± √32 ] / 2
x = [8 ± 4√2 ] / 2
x = 4 + 2√2 (taking + ve value for x

Question 3
log [(x + 3)(x) ] = log 7
(x + 3) (x) = 7
x² + 3x - 7 = 0
x = [- 3 + √37 ] / 2 (taking + ve value)

Answer 4

To solve these log questions, you need to know the basic log identities...you can find these on wiki:

http://nostalgia.wikipedia.org/wiki/Logarithm/Identities

For example, the first question can be answered by combining the log terms on the left side of the equation:

log(x+2)+log(x-2)= log((x+2)*(x-2))

Therefore, we know that (x+2)*(x-2)=7 by comparison of each side of the equation. The other two question can be solved by using some of the identities listed on the wiki site. Good luck!

Answer 5

1.
By the law of logs this means that u have to find x when
log (x-2)(x+2)=log 7
log (x^2-4)= log 7
x^2-4=7
x^2=11
x= root 11