How to know the exponent of a given function to be equal or greater than a given number

Question

I have the following function: $f(x)=(0.9)^x$. I need to calculate for which $x$ it is true that $f(x) > 0.5$. How could I calculate it? Can I use a logarithm base 0.9?

$$ \log_{0.9}0.5=x $$

If so, how could I use it to do the calculation?

Answer 1

Write $0.9=9/10$ so that you want $x$ such that $$\left(\frac{9}{10}\right)^x>\frac{1}{2}\iff 2(9^x)>10^x\\\iff \log(2)+x\log(9)>x\log(10)\\\iff x<\frac{\log(2)}{\log(10)-\log(9)}$$ Note: the choice of log base doesn't matter here. It could be $e$ or $10$ as are fairly common, although you could choose base $21$ if you want or whatever.

As a note, recall that we have $$\log(b^x)=x\log(b),\qquad\log(ab)=\log(a)+\log(b)$$