Logarithmic Functions?

Question

Solve each equation for x to the nearest hundredth.

1) log (x -3) + log x = log 7

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

explain please

Answer 1

Using these formulae:

1) log ab = log a+ log b
2) log a = log b implies a = b


1) log (x -3) + log x = log 7
log (x-3)x = log 7
x(x-3) = 7
x² - 3x - 7 = 0
Solving the quadratic equation by quadratic formula:
x = 4.54 or x = -1.54

2) log (x +2) + log (x -2) = 1
log (x+2)(x-2) = log 10
x² - 4 = 10
x² = 14
x = 3.74 or -3.74

Answer 2

Both of these use the fact that log(a) + log(b) = log(a*b).
1. log(x - 3) + log x = log 7
log x(x + 3) = log 7
x(x + 3) = 7
Solve using the quadratic formula.

2. log(x + 2) + log(x - 2) = 1
log(x + 2)(x - 2) = 1
Assuming you are using logs with base 10 then
(x + 2)(x - 2) = 10
When you multiply out the brackets you will see that this one doesn't even need the quadratic formula.

Answer 3

Question 1
(x - 3).(x) = 7
x² - 3x - 7 = 0
x = (3±√37) / 2
x = 4.54, x = - 1.54

Question 2
Take logs as being to base ten
log(x + 2) + log(x - 2) = 1
log ((x + 2).(x - 2)) = 1
(x + 2).(x - 2) = 10
x² - 4 = 10
x² = 14
x = ± √14
x = ± 3.74