Number of positive solutions to $\log_{10}(x+\pi)=\log_{10}x+\log_{10}\pi$

Question

For $x>0$, how many solutions does this equation have? $$\log_{10}(x+\pi)=\log_{10}x+\log_{10}\pi$$ I just don't know where to start. I'm not very familiar to log functions and log problems in general.

Answer 1

Rewrite the right-hand side using the property $\log ab=\log a+\log b$: $$\log_{10}(\pi+x)=\log_{10}\pi x$$ Exponentiate both sides by 10: $$\pi+x=\pi x$$ This is a linear equation in $x$ and therefore has only one solution. $$x=\frac\pi{\pi-1}$$

Answer 2

Try to exponentiate using the laws of logarithm resp. exponential, $$ x+π=x\cdot π $$ has an easy solution.

Answer 3

$$\log(x+\pi)=\log(x)+\log(\pi)=\log(x\pi)$$

$$x+\pi=x\pi$$

$$x=\frac\pi{\pi-1}$$