Find the exact solution of the exponential equation in terms

tugmiddelc0 2021-11-21 Answered
Find the exact solution of the exponential equation in terms of logarithms.
Use a calculator to find an approximation to the solution rounded to six decimal places.
I keep getting a negative log answer and decimal...i don't think that is correct...can i get some help please.
\(\displaystyle{8}^{{{1}-{x}}}={9}\)

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Expert Answer

Feas1981
Answered 2021-11-22 Author has 1700 answers
Our aim is to find the exact the exact solution of the exponential equation
\(\displaystyle{8}^{{{1}-{x}}}={9}-{\left({i}\right)}\) in terms of logarithms.
Taking the logarithmic of equation−(i) both sides we have:−
\(\displaystyle{\ln{{\left({8}^{{{1}-{x}}}\right)}}}={\ln{{\left({9}\right)}}}-{\left({i}{i}\right)}\)
Now, using Logarithmic Power Rule in L.H.S of equation (ii), we have:−
\(\displaystyle{{\log}_{{b}}{\left({x}^{{y}}\right)}}={y}{{\log}_{{b}}{\left({x}\right)}}\rightarrow{\left[\text{Logarithmic Power Rule}\right]}\)
\(\displaystyle\Rightarrow{\left({1}−{x}\right)}{\ln{{\left({8}\right)}}}={\ln{{\left({9}\right)}}}−{\left({i}{i}{i}\right)}\)
Applying Distributive Property in L.H.S. of equation (iii), we have:−
\(\displaystyle\Rightarrow{\ln{{\left({8}\right)}}}−{x}{\ln{{\left({8}\right)}}}={\ln{{\left({9}\right)}}}\)
\(\displaystyle\Rightarrow{x}{\ln{{\left({8}\right)}}}={\ln{{\left({9}\right)}}}−{\ln{{\left({8}\right)}}}\)
\(\displaystyle\Rightarrow{x}={\frac{{{\ln{{\left({9}\right)}}}-{\ln{{\left({8}\right)}}}}}{{{\ln{{\left({9}\right)}}}}}}\)
\(\displaystyle\Rightarrow{x}={\frac{{{2.197224}-{2.079441}}}{{{2.079441}}}}\)
\(\displaystyle\Rightarrow{x}={\frac{{{0.117783}}}{{{2.079441}}}}\)
\(\displaystyle\Rightarrow{x}={0.056642}\ \text{ is the exact solution of exponential equation }\ {8}^{{{1}-{x}}}={9}\ \text{ upto 6 decimal places.}\)
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Ourst1977
Answered 2021-11-23 Author has 458 answers
Thank you very much for the solution, I have been looking for it for a long time and could not find
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