For f(x) = log2 (x + 5) use the change of base formula to approximate the y-intercept.
y = log_2(x+5)
when x=0,
y = log_2(5) = log5/log2 = ___
To approximate the y-intercept of the function f(x) = log2 (x + 5) using the change of base formula, we first need to rewrite the function using a different base.
The change of base formula states that any logarithm can be expressed in terms of a different base logarithm. In this case, we can rewrite the logarithm with base 2 as a logarithm with base 10, which allows us to calculate it numerically:
log2(x) = log10(x) / log10(2)
By applying the change of base formula to our function f(x), we get:
f(x) = log2(x + 5) = log10(x + 5) / log10(2)
Now, to approximate the y-intercept, we need to find the value of x where f(x) equals 0. Since the y-intercept corresponds to the point where the function crosses the y-axis (x = 0), we can substitute x = 0 into the equation:
f(0) = log10(0 + 5) / log10(2)
However, taking the logarithm of 0 is undefined, so we cannot directly substitute x = 0 into the equation. Instead, we need to find a value close to 0 and substitute it. Let's use x = 0.001 as an approximation:
f(0.001) = log10(0.001 + 5) / log10(2)
Using a scientific calculator or math software, evaluate log10(0.001 + 5) and log10(2) separately, and then divide them to get the value of f(0.001). This will give you an approximation of the y-intercept of the function f(x) = log2(x + 5).