I noticed that after cutting through $(1,0)$ the function $y=\log_{10}(x)$ is steeper .

$x=10^y$ , $x=20^y$

From this , the $x$ values must be the same, therefore the $y$ value on $20^y$ will be smaller .

However, now , let's compare


$$\log_{0.9}(y) = y$$

From the first example , we learnt that the bigger the base , the less steep the graph will be after $(1,0)$. Why is this not the case for these $2$ functions?

$0.9$ is a bigger number than $0.5$, but why does the '$0.9$' graph continue to be steeper than the '$0.5$' graph after $(1,0)$?


Note that the function $0.5^x$ is "steeper" than $0.9^x$, because $0.5^x$ decreases more rapidly. Their inverse functions are negative for $x>1$, so therefore the function that the lesser function will be more steep. Also, consider the derivatives of $\log_{0.9}x$ and $\log_{0.5}x$: $(\log_{0.9}x)' = \frac{1}{x\ln{0.9}}$, and $(\log_{0.5}x)' = \frac{1}{x\ln{0.5}}$. As $\frac{1}{\ln{0.9}} \approx -9.491$, and $\frac{1}{\ln{0.5}} \approx -1.443$, we see from here that the function $\log_{0.9}x$ will decrease more intensely than $\log_{0.5}x$.

To see this more clearly, just remember that things "reverse" when the base of the logarithm becomes lesser than $1$, compared to when it was greater than $1$.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.