Question

45 pointsSolve the logarithmic equation below. All work must be shown to earn full credit and

Answer 1

Logarithm properties

We know that the substraction of two logarithm of the same base is related to a division:

$\log _460-\log _44=\log _4(\frac{60}{4})$

Since 60/4 = 15, then

$\log _4(k^2+2k)=\log _415$

Then, the expressions in the parenthesis are equal:

k² + 2k = 15

Factoring the expression

Now, we can solve for k:

k² + 2k = 15

↓ substracting 15 both sides

k² + 2k - 15 = 0

Since

5 · (-3) = -15 [third term]

and

5 - 3 = 2 [second term]

we are going to use 5 and -3 to factor the expression:

k² + 2k - 15 = (k -3) (k +5) = 0

We want to find what values should have k so

(k -3) (k +5) = 0

if k -3 = 0 or if k +5 = 0, the expression will be 0

So

k - 3 = 0 → k = 3

k +5 = 0 → k = -5

Answer: k = 3 or k = -5