Fixing the Equation: log₆(x) + log₆(x – 5) = 2
This text will information you thru fixing the logarithmic equation log₆(x) + log₆(x – 5) = 2. We’ll break down the method step-by-step, making it straightforward to grasp even for those who’re new to logarithms.
Understanding the Drawback
The equation log₆(x) + log₆(x – 5) = 2 includes the logarithm base 6. Keep in mind that a logarithm is basically the inverse of an exponent. In easier phrases, log₆(x) asks “6 to what energy equals x?”.
Making use of Logarithmic Properties
To unravel this equation, we have to make the most of a key logarithmic property: the product rule. This rule states that logₐ(b) + logₐ(c) = logₐ(bc). Making use of this to our equation:
log₆(x) + log₆(x – 5) = log₆[x(x – 5)] = 2
Now our equation is simplified to:
log₆[x(x – 5)] = 2
Changing to Exponential Kind
To eliminate the logarithm, we’ll convert the equation from logarithmic type to exponential type. Keep in mind that logₐ(b) = c is equal to aᶜ = b. Subsequently:
6² = x(x – 5)
Fixing the Quadratic Equation
Increasing the equation, we get a quadratic equation:
36 = x² – 5x
Rearranging to straightforward quadratic type (ax² + bx + c = 0):
x² – 5x – 36 = 0
Now we will remedy this quadratic equation utilizing factoring, the quadratic components, or finishing the sq.. Factoring is the simplest methodology on this case:
(x – 9)(x + 4) = 0
This offers us two potential options:
x = 9 or x = -4
Checking for Extraneous Options
It is essential to verify if each options are legitimate. Keep in mind that the argument of a logarithm (the worth contained in the parentheses) should at all times be constructive.
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x = 9: log₆(9) + log₆(9 – 5) = log₆(9) + log₆(4). Each arguments are constructive, so this answer is legitimate.
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x = -4: log₆(-4) is undefined since you can not take the logarithm of a unfavorable quantity. Subsequently, x = -4 is an extraneous answer and should be discarded.
The Resolution
Subsequently, the one legitimate answer to the equation log₆(x) + log₆(x – 5) = 2 is x = 9.
Verifying the Resolution
Let’s confirm our answer by plugging x = 9 again into the unique equation:
log₆(9) + log₆(9 – 5) = log₆(9) + log₆(4) ≈ 1.2265 + 0.7735 = 2
The equation holds true, confirming our answer.
This detailed clarification ought to provide help to perceive the best way to remedy logarithmic equations successfully. Keep in mind to at all times verify for extraneous options to make sure your reply is legitimate. When you encounter extra complicated logarithmic equations, bear in mind to make the most of different logarithmic properties resembling the ability rule and quotient rule as wanted.
