Obtain graph from a logarithmic function

Question

Consider the function

$$f(x)=-2\log(x-10)$$

How can I obtain its graph from transforming the function

$$g(x)=10^x\text{ ?}$$

I guess with $\log$ they mean the logarithm to base $10$. So what I tried to do is

$$10^{-2}\cdot(x-10)=g(f(x))$$

So I would first shift graph of $id(x)=x$ to the right by $10$ and then adjust its slope by $10^{-2}$ ?

Finally I take the logarithm of that?

Answer 1

Recognize that $$g(x)=10^x\iff g^{-1}(x)=\log x$$ and as such, we can write $$f(x)=-2g^{-1}(x-10).$$

Therefore, from the graph of $g(x)$, to get $f(x)$ we can perform the following operations in order:

• Reflect across the line $y=x$
• Reflect across the line $y=0$
• Expand vertically by a factor of $2$
• Translate $10$ units to the right