Solve for x: log (x+5) = 2 - log (2x)

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

The equation to be solved is log (x+5) = 2- log (2x)

log (x+5) = 2- log (2x)

log (x+5)  + log 2x = 2

use the property that log a + log b = log a*b

=> log (x + 5)*2x = 2

=> 2x^2 + 10x = 100

=> x^2 + 5x - 50 = 0

=> x^2 + 10x - 5x - 50 = 0

=> x(x + 10) - 5(x + 10) = 0

=> (x - 5)(x + 10) = 0

=> x = 5 and x = -10

As the log of negative numbers is not defined, we only have x = 5

The solution of the equation is x = 5

hala718's profile pic

hala718 | High School Teacher | (Level 1) Educator Emeritus

Posted on

log (x+5) = 2- log (2x).

I believe that you need to determine the values of x that satisfies the equation.

We will use logarithm properties to solve for x.

First we will add log 2x to both sides.

==> log (x+ 5) + log (2x) = 2

Now we know that log a + log b= log a*b.

==> log (x+5)*2x = 2

==> log ( 2x^2 +10x) = 2

Now we will rewrite into the exponent form.

==> 2x^2 + 10x = 10^2

==> 2x^2 + 10x = 100

==> 2x^2 + 10 x -100 = 0

Now we will divide by 2.

==> x^2 + 5x -50 = 0

Now we will factor.

==> (x+10) (x-5) = 0

==> x1= -10 ==> Not valid because log ca not be negative.)

==> x2= 5

Let us check .

==> log (5+5) = 2 - log 2*5

==> log 10 = 2- log 10

==> 1 = 2-1

==> 1 = 1

Then the answer is valid

==> Then the answer is x= 5.

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