Answer:
x = -4
Step-by-step explanation:
To solve the logarithmic equation log₄(x+8) + log₄(x+5) = 1, we can simplify it by using the properties of logarithms.
First, remember that adding two logs with the same base is the same as taking the log of the product of their arguments. So:
log₄(x+8) + log₄(x+5) = log₄((x+8)(x+5))
Now we have:
log₄((x+8)(x+5)) = 1
The next step is to understand that log₄ of something equals 1 means that "something" is 4¹ (because the base is 4 and any number raised to the power of 1 is itself). So:
(x+8)(x+5) = 4¹
Which simplifies to:
(x+8)(x+5) = 4
Now we just need to solve this quadratic equation. First, expand the left side:
x² + 5x + 8x + 40 = 4
Combine like terms:
x² + 13x + 40 = 4
Now, subtract 4 from both sides to set the equation to zero:
x² + 13x + 36 = 0
This is a quadratic equation that we can factor:
(x + 9)(x + 4) = 0
From the factored form, we have two possible solutions:
x + 9 = 0 or x + 4 = 0
Solving for x gives us:
x = -9 or x = -4
However, we need to check these solutions to make sure they make sense in the context of the original logarithmic equation, because you can't take the log of a negative number.
If x = -9, then log₄(x+8) and log₄(x+5) would be log₄(-1) and log₄(-4), which are undefined in real numbers.
If x = -4, then log₄(x+8) and log₄(x+5) would be log₄(4) and log₄(1). Since log₄(4) = 1 (because 4 is the base) and log₄(1) = 0 (any log base of 1 is 0), these are defined and the original equation holds true.
So, the exact solution is:
x = -4
Hope this helps!
