Logarithmic equation?

Question

Please help with the equation below:

log[base 10](x+7) - log[base 10](x-7)=1

Thank you!

Answer 1

Solution :
x=77/9
Here's how i got it:

The equation is:
Log(x+7) - Log(x-7) = 1

1 can be written as: Log (base10) 10
beacause Log (base n) n = 1

So, the equation now is:
Log(x+7) - Log(x-7) = Log10
Log [(x+7)/(x-7)] = Log10

Now, removing Log from both sides,
(x+7)/(x-7) = 10
x+7 = 10(x-7)
x+7 = 10x - 70
9x = 77
x = 77/9

Answer 2

Dont forget to study where these logarith exist. IF YOU DONT STUDY THE EXPRESSIONS EXISTENCE, YOUR EXERCISE COULD EVEN BE COMPLETELY WRONG. None of the former answerers considered this and this is a too important flaw. Pay lots of attention to this, please.

log (x+7) exists iff x >-7

And log (x-7) exists iff x>7

But both have to exist at the same time, so, x must be greater than 7. Any solution that you get and that is less than or equals to 7 must be discharted

You may do this too:

log(x+7) = log (10-7) + 1

And, if you remember, 1= log 10

So, you can apply logarith properties here:

log (x+7) = log (x-7) + log 10

log (x+7) = log 10 (x-7)

Hence x+7 = 10(x-7)

x+7 = 10x - 70

And 9x = 77

x = 77/9, which is greater than 7.

Instead of studying your expressions existence, you can just plug your value in it and see if all is OK

Take care

Ana

Answer 3

Assuming log means log base 10

log(x+7) - log(x-7) = 1 leads to (definition)

log((x+7)/(x-7)) = log(10)

Thus

x + 7 = 10 (x - 7)

9x = 77

x = 77/9

Answer 4

using the subtraction rule,
log[base 10] (x+7)/(x-1) = 1, so by definition of log,
(x+7)/(x-7) = 10^1 = 10
x + 7 = 10(x - 7)
x + 7 = 10x - 70
77 = 9x
8 5/9 = x

Answer 5

log[x+7] - log[x-7] = 1 = log[10]
log[x+7] - log[x-7] - log[10] = 0
log[(x+7)/(x-7) = 1
Take 10 to both sides:
(x+7)/(x-7) = 10
x+7 = 10x - 70
9x = 77
x = 77/9

Answer 6

First, base 10 is called the common logarithm, and most of us just write "log" for it.

Use the difference rule: log(x+7) - log(x-7) = log((x+7)/(x-7))
So: log((x+7)/(x-7)) = 1
Then use the exponent rule of equations: 10^log((x+7)/(x-7)) = 10^1
So: (x+7)/(x-7) = 10

I'll let you solve it from there.

Answer 7

log(10)(x + 7) - log(10)(x - 7) = 1
log(10)((x + 7)/(x - 7)) = 1
10^1 = (x + 7)/(x - 7)
(x + 7)/(x - 7) = 10
x + 7 = 10x - 70
-9x = -77
x = (77/9)

ANS : x = (77/9)

Answer 8

x=77/9