f(x) = log₂(x+2)-2
Translate base-2 log left 2, down 2
First, let’s understand the basics of the logarithm function. The logarithm function, denoted as log, is the inverse of an exponential function. In other words, if we have an equation of the form y = a^x, the inverse function would be x = log_a(y).
Now, let’s analyze the given function f(x) = log₂(x+2)-2.
The function has two parts – a logarithmic function and a constant.
The logarithmic function is log₂(x+2), which means that it is a base-2 logarithm of the quantity (x+2).
The constant is -2, which means that the entire function is shifted downward by 2 units.
When we combine the logarithmic function and the constant, we get the function f(x) = log₂(x+2)-2.
To evaluate this function, we can pick a value of x and substitute it into the equation. For example, if we want to evaluate f(3), we substitute the value of x=3 into the equation:
f(3) = log₂(3+2)-2
f(3) = log₂(5)-2
We can use a calculator or the change of base formula to simplify the logarithmic expression:
f(3) = log₂(5)-2
f(3) = (logₑ(5))/(logₑ(2))-2 … using the change of base formula
f(3) ≈ 0.32193 – 2
f(3) ≈ -1.67807
Therefore, f(3) ≈ -1.67807.
We can use the same steps to evaluate the function for any value of x.
Note that the domain of the function is restricted to x > -2, since we cannot take the logarithm of a negative number.
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