# Solve for x 1200(1.03)^x=800(1.08)^x

Question
Logarithms
Solve for x $$\displaystyle{1200}{\left({1.03}\right)}^{{x}}={800}{\left({1.08}\right)}^{{x}}$$

2020-11-09
$$\displaystyle{1200}{\left({1.03}^{{x}}\right)}={800}{\left({1.08}^{{x}}\right)}$$
divide by 800 on both sides
divide by $$\displaystyle{1.03}^{{x}}$$ on both sides
$$\displaystyle\frac{{1200}}{{800}}=\frac{{{1.08}^{{x}}}}{{{1.03}^{{x}}}}$$
$$\displaystyle\frac{{3}}{{2}}={\left(\frac{{1.08}}{{1.03}}\right)}^{{x}}$$
$$\displaystyle{\ln{{\left(\frac{{3}}{{2}}\right)}}}={\ln{{\left({\left(\frac{{1.08}}{{1.03}}\right)}^{{x}}\right)}}}$$
$$\displaystyle{\ln{{\left(\frac{{3}}{{2}}\right)}}}={x}\cdot{\ln{{\left(\frac{{1.08}}{{1.03}}\right)}}}$$
Exact: $$\displaystyle{x}=\frac{{\ln{{\left(\frac{{3}}{{2}}\right)}}}}{{\ln{{\left(\frac{{1.08}}{{1.03}}\right)}}}}$$
Decimal, approximate: x = about 8.554

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