£4500 is invested at 3.2% compound interest per year. how many years will it take for the investment to exceed £5000.

$$\\4500(1+0.032)^n>5000\\\\
(1+0.032)^n>\frac{10}{9}\\\\
log(1.032)^n>log\frac{10}{9}\\\\
nlog(1.032)>log\frac{10}{9}\\\\
n>\frac{log\frac{10}{9}}{log(1.032)}\\\\$$

 

$${\frac{{log}_{10}\left({\frac{{\mathtt{10}}}{{\mathtt{9}}}}\right)}{{log}_{10}\left({\mathtt{1.032}}\right)}} = {\mathtt{3.344\: \!919\: \!817\: \!058\: \!767\: \!9}}$$

 

$$\\n>3.3 years\\
n=4\;\;years$$

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