Solve for the following exponential equations. Use the natural logarithm in your

sagnuhh 2021-09-30 Answered
Solve for the following exponential equations. Use the natural logarithm in your answer(where applicable) for full credit. Use rules for exponents, factor and simplify.
\(\displaystyle{10}^{{{6}-{3}{x}}}={18}\)

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Expert Answer

au4gsf
Answered 2021-10-01 Author has 10142 answers
Given that:
The equation \(\displaystyle{10}^{{{6}-{3}{x}}}-{18}\).
By using,
Natural Logarithm rule:
Exponent rule,
\(\displaystyle{\ln{{\left({x}^{{y}}\right)}}}={y}{\ln{{\left({x}\right)}}}\)
To solve the given equation.
By using the above formula,
Taking natural log on both side,
\(\displaystyle{\ln{{\left({10}^{{{6}-{3}{x}}}\right)}}}={\ln{{\left({18}\right)}}}\)
\(\displaystyle{\left({6}-{3}{x}\right)}{\ln{{\left({10}\right)}}}={\ln{{\left({18}\right)}}}\)
Divide on both side by \(\displaystyle{\ln{{\left({10}\right)}}}\)
To get,
\(\displaystyle{6}-{3}{x}={\frac{{{\ln{{\left({18}\right)}}}}}{{{\ln{{\left({10}\right)}}}}}}\)
\(\displaystyle-{3}{x}={\frac{{{\ln{{\left({18}\right)}}}}}{{{\ln{{\left({10}\right)}}}}}}-{6}\)
\(\displaystyle{x}={\frac{{{1}}}{{-{3}}}}{\frac{{{\ln{{\left({18}\right)}}}}}{{{\ln{{\left({10}\right)}}}}}}+{3}\)
\(\displaystyle{x}={\frac{{{1}}}{{-{3}}}}{\left({1.2553}\right)}+{3}\)
\(\displaystyle={\frac{{{1.2553}}}{{-{3}}}}+{3}\)
\(\displaystyle=-{0.4184}+{3}\)
\(\displaystyle={2.5816}\)
To get,
\(\displaystyle{x}={2.5816}\)
Therefore, \(\displaystyle{x}={2.5816}\)
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