find x: log x + log 10x = 3; 3(10^(x+4) - 3) = 9.?

Answer 1

log(x) + log(10x) = 3
log(x*10x) = 3
log(10x^2) = 3
10x^2 = 10^3
10x^2 = 1000
Divide both sides by 10
x^2 = 100
Take the square root of each side
x = -10 or x = 10
Remember that you can only take the square root of positive numbers.
Answer: x = 10

3(10^(x+4) - 3) = 9
Divide both sides by 3
10^(x+4) - 3 = 3
Add 3 to each side
10^(x+4) = 6
Take the log of each side
log(10^(x+4)) = log(6)
x + 4 = log(6)
Subtract 4 from each side x = log(6) - 4

Answer 2

log x + log 10x = 3
log x + (log 10 + log x) = 3
2 log x + log 10 = 3
2 log x + 1 = 3
2 log x = 3 - 1
2 log x = 2
log x = 2 / 2
log x = 1 ------> x = 10^1 = 10.

Substitute it into the equation. It works.

#2
3(10^(x+4) - 3) = 9
3(10^(x+4) - 3) / 3 = 9 / 3
(10^(x+4) - 3) = 3
10^(x + 4) - 3 + 3 = 3 + 3
10^(x + 4) = 6
log [10^(x + 4)] = log 6
(x + 4) = log 6
(x + 4) - 4 = (log 6) - 4
x = (log 6) - 4 ≈ -3.2218

Substitute this one in also. It works as well.

Answer 3

log x + log 10x = 3
log x(10x) = 3 (product of log)
log 10x^2 = 3 (distributive)
10x^2 = 1000 (rewrite in exponential)
x^2 = 100
x= +/- 10
you can't have a negative answer in this problem, therefor, only 10 is the answer

3(10^(x+4) - 3) = 9
10^(x+4) -3 = 3
10^(x+4) = 6
(x+4) log10 = log6
x+4 = log6
x = log6 - 4
x= -3.221

Answer 4

1) Assume log(x) = log base 10

2) 3 = log(x) + log(10x) = log(x) + log(10) + log(x)= 2log(x) + 1

3) log(x) = 1;

4) x = 10;

--------------------------------

3(10^(x+4)-3) = 9
10^(x+4) = 6
x+4 = log(6);
x = log(6) - 4

Answer 5

1) logx+log10x=3
>>logx+log10+logx=3
>>1+2logx=3
>>logx=1
>>x=10

2)3(10^(x+4) - 3) = 9
>>10^(x+4)=6
>>x+4=log6
>>x+4=log2+log3
>>x+4=0.3010+0.4771
>>x+4=0.7781
>>x= -3.2219