Question

f(x)=log2(x-5)+2 help Find x-intercept and end behavior

Answer 1

To find the x intercept, we need to find the point where f(x)=0

$0=\log 2(x-5)+2$

And we solve this equation for x:

Substract 2 from both sides:

$-2=\log 2(x-5)$

Now, we do the following to eliminate the log:

$10^{-2}=10^{\log 2(x-5)}$

We make the expression we had, the exponents of the new expression of base 10. This so that in the right side the 10 and the log will cancel:

$10^{-2}=2(x-5)$

And now we use distributive property on the right side:

$10^{-2}=2x-10$

we can express the left side as 0.01:

$0.01=2x-10$

Add 10 to both sides:

$\begin{gathered} 0.01+10=2x-10+10 \\ 10.01=2x \end{gathered}$

We divide both sides by 2:

$\begin{gathered} \frac{10.01}{2}=\frac{2x}{2} \\ 5.005=x \end{gathered}$

The x intercept is x=5.005

To find the end behavior we can use the graph of the function:

The function starts at x=5.005 and as x grows, the graph of the function stays close to the x-axis around the value of y=4.