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

Question
Logarithms
asked 2020-11-08
Solve for x \(\displaystyle{1200}{\left({1.03}\right)}^{{x}}={800}{\left({1.08}\right)}^{{x}}\)

Answers (1)

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
0

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