logarithm sum-product propertydiscuss it on example

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hala718's profile pic

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

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The property is:

log (a) + log (b) = log (ab)

The bases of the logarithm must be equal.

Let us solve an example.

Given the logarithm equation:

log (x-2) + log x = log 4

We need to solve for x.

We will use the sum property to solve.

==> log (x-2) + log x = log (x+2)*2x = log 4

==> log (x^2 - 4x ) = log 4

Now that the logs are equal, then the bases are equal too.

==> x^2 - 4x = 4

==> x^2 - 4x - 4 = 0

==> x1= ( 4 + sqrt(16+16) / 2 = (4+4sqrt2)/2 = 2+ 2sqrt2

==> x1= 2+2sqrt2

==> x2= 2-2sqrt2  ( Not valid because x2 < 0)

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

Let's analyze the logarithmic equation:

lg (8x+9) + lgx = 1 + lg(x^2 - 1)

Before solving it, we'll impose the constraints of existence of logarithms:

8x+9 > 0

x > -9/8

x^2 - 1 > 0

(x-1)(x+1) > 0

x > 1

x > -1

The interval of admissible values for x is (1 ; +infinite).

Now, we'll solve the equation:

lg (8x+9) + lgx = 1 + lg(x^2 - 1)

We'll write 1 = lg 10 and we'll apply the product rule of logarithms both sides:

log a + log b  = log (a*b)

lg (8x+9) + lgx = lg [x*(8x+9)]

lg10 + lg(x^2 - 1) = lg 10(x^2 - 1)

lg [x*(8x+9)] = lg 10(x^2 - 1)

Since the bases are matching, we'll apply one to one property:

 [x*(8x+9)] = 10(x^2 - 1)

We'll remove the brackets:

8x^2 + 9x = 10x^2 - 10

We'll subtract 8x^2 + 9x and we'll use symmetric property:

2x^2 - 9x - 10 = 0

We'll apply quadratic formula:

x1 = [9 + sqrt(81 + 80)]/4

x1 = (9+sqrt161)/4

x2 = (9-sqrt161)/4

Since the value of x2 doesn't belong to the interval of admissible values, we'll reject x2.

The only valid solution of the equation is x1 = (9+sqrt161)/4

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