Which of the following shows the true solution to the logarithmic equation below ? log(x) + log(x + 5) = log(6x + 12)
x = -3
x = 4
x = -3 and x = 4
x = -3 and x = -4

Solution:

Given, the logarithmic equation is log(x) + log(x + 5) = log(6x + 12)

We have to find the true solution to the logarithmic equation.

By using logarithmic property,

\(log\, a+log\, b=log\, (a\times b)\)

So, log(x) + log(x + 5) = log(x(x + 5))

= log(x² + 5x)

Now, log(x² + 5x) = log(6x + 12)

Cancelling log on both sides,

x² + 5x = 6x + 12

x² + 5x - 6x - 12 = 0

x² - x - 12 = 0

On factoring,

x² - 4x + 3x - 12 = 0

x(x - 4) + 3(x - 4) = 0

(x + 3)(x - 4) = 0

x + 3 = 0 ⇒ x = -3

x - 4 = 0 ⇒ x = 4

Since x cannot be negative, the only solution is x = 4.

Therefore, the true solution is x = 4.

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Which of the following shows the true solution to the logarithmic equation below ? log(x) + log(x + 5) = log(6x + 12)

Summary:

The true solution to the logarithmic equation log(x) + log(x + 5) = log(6x + 12) is x = 4.