logarithms?

Question

If log(x+54) - log(x+2) = 2 - log(x), then what is x?

Answer 1

To solve logarithmic equation, these three log properties are likely to come into play:

1) log[base b](ac) = log[base b](a) + log[base b](c)
2) log[base b](a/c) = log[base b](a) - log[base b](c)
3) log[base b](a^c) = c * log[base b](a)

Also, remember that log[base b](a) = c in exponential form is
b^c = a

Now, let's begin

log(x + 54) - log(x + 2) = 2 - log(x)

First, apply log property #2 to the left hand side.

log[ (x + 54)/(x + 2) ] = 2 - log(x)

Move the -log(x) over to the left hand side.

log[ (x + 54)/(x + 2) ] + log(x) = 2

Apply log property #1.

log[x(x + 54)/(x + 2)] = 2

Now, convert this to exponential form.

x(x + 54)/(x + 2) = 10^2

Multiply both sides by (x + 2), to get rid of the fraction, to get

x(x + 54) = 10^2 (x + 2)

Simplify.

x^2 + 54x = 100(x + 2)
x^2 + 54x = 100x + 200
x^2 - 46x - 200 = 0

This factors into

(x - 50) (x + 4) = 0
So we get a solution of x = {50, -4}

BUT WAIT! We CANNOT assume both of these values work, because we cannot take the log of a negative number. What we have to do is TEST these values by plugging them back into the original equation, log(x + 54) - log(x + 2) = 2 - log(x)

Test x = 50: log(50 + 54) - log(50 + 2) = 2 - log(50).
We're not taking the log of a negative number, so this checks out.

Test x = -4; we can immediately see that on the right hand side, we're taking log(-4), so we REJECT this solution.

Therefore, x = 50 is the only solution.

Answer 2

log (x+54/x+2) = 2 - log x
=>log (x+54/x+2) = log 100 - log x

=>log (x+54/x+2) = log (100/x)

=>x+54 / x+2 = 100 / x

=>x^2 + 54x = 100x + 200

=>x^2 - 46x - 200 = 0

=>(x - 50)(x + 4) = 0

=>x = 50 or x = -4
Log of non positive numbers is not defined.

So, x = 50

Answer 3

log(x+54) - log(x+2) = 2 - log(x)
log(x+54)/log(x+2) = log 100/logx
x+54/x+2 = 100/x
x^2 + 54x = 100x + 200
x^2 - 46x - 200 = 0
(x - 50)(x + 4) = 0
x = 50 or -4 (rejected since log cannot be in -ve form)

Answer 4

log (x+54/x+2) = 2 - log x
log (x+54/x+2) = log 100 - log x
log (x+54/x+2) = log (100/x)
x+54 / x+2 = 100 / x
x^2 + 54x = 100x + 200
x^2 - 46x - 200 = 0
(x - 50)(x + 4) = 0

x = 50 or x = -4
Log of non positive numbers is not defined.

So, x = 50

Answer 5

If log(x+54) - log(x+2) = 2 - log(x)
log x(x+54)/(x+2)=2
x(x+54)/(x+2)=100
x^2+54x=100x+200
x^2-46x-200=0
(x-50)(x+4)=0
x=50, -4

Answer 6

log (x+54/x+2) = 2 - log x
log (x+54/x+2) = log 100 - log x
log (x+54/x+2) = log (100/x)
x+54 / x+2 = 100 / x
x^2 + 54x = 100x + 200
x^2 - 46x - 200 = 0
(x - 50)(x + 4) = 0

x = 50 or x = -4
Log of non positive numbers is not defined.

So, x = 50

Answer 7

log(x + 54) - log(x + 2) = 2 - log(x)
log[x(x + 54)/(x + 2)] = 2
x^2 + 54x = 100x + 200
x^2 - 46x - 200 = 0
(x - 50)(x + 4) = 0
x = -4, 50

Here's how -4 can be a solution:
log(x + 54) = 2 - log(x) + log(x + 2)
log(x + 54) = log[(100)(x + 2)/x]
log(50) = l0g(100(-2)/(-4) = log(100(2)/4)

Answer 8

log(x + 54) - log(x + 2) = 2 - log(x)
so, log( (x + 54) / (x + 2) ) = log 100 - log(x)
so, (x + 54) / (x + 2) = 100/x
so, x^2 + 54x = 100x + 200
so, x^2 - 46x - 200 = 0
so, x^2 - 50x + 4x - 200 = 0
so, (x - 50) (x + 4) = 0
as x cannot be negative, x = 50.

Answer 9

Q: log(x+54)-log(x+2)=2-log(x)

ans: Add log(x) on both sides

you get=> log(x+54)-log(x+2)+log(x)=2

now use this law: logm-logn=log(m/n)

log[(x+54)/(x+2)]+log(x)=2


now use this law: logm+logn=log(m*n)

log[[(x+54)/(x+2)]*x]=2

as u know that log100=2

log[[(x+54)/(x+2)]*x]=log100

by eliminating log from both sides u will get;

[ [(x+54)/(x+2)]*x]=100

now solve it:
x^2+54x=100*(x+2)
x^2+54x-100x-200=0
x^2-46x-200=0
x^2-50x+4x-200=0
x(x-50)+4(x-50)=0
(x-50)(x+4)=0
u will get:
x=50
x= - 4
since the log of negative values does not exist so we can discard x=-4 to avoid this problem now your final answer is:

x=50

Answer 10

50