Question

Exponential and Logarithmic Equations solve the equation e^{3x/4}=10

Decimals
ANSWERED
asked 2021-08-11
Exponential and Logarithmic Equations solve the equation. Find the exact solution if possible. otherwise, use a calculator to approximate to two decimals.
\(\displaystyle{e}^{{{3}\frac{{x}}{{4}}}}={10}\)

Answers (1)

2021-08-12
Step 1
Law of logarithm:
Consider m to be a positive number, and \(\displaystyle{m}\ne{q}{1}\)
Again consider M and N to be any real numbers with \(\displaystyle{M}{>}{0}\) and \(\displaystyle{N}{>}{0}\)
The logarithm of power of a number is equal to the exponen times the logarithm of the number as,
\(\displaystyle{{\log}_{{{m}}}{\left({M}^{{{n}}}\right)}}={N}{{\log}_{{{m}}}{M}}\)
Step 2 The given exponential equation is,
1) \(\displaystyle{e}^{{{\frac{{{3}{x}}}{{{4}}}}}}={10}\)
Take logarithm on both side of the equation (1).
2) \(\displaystyle{\ln{\ }}{e}^{{{\frac{{{3}{x}}}{{{4}}}}}}={\ln{{10}}}\)
Therefore, the equation (2) can be written as,
\(\displaystyle{\frac{{{3}{x}}}{{{4}}}}{\ln{\ }}{e}={\ln{{10}}}\)
\(\displaystyle{\frac{{{3}{x}}}{{{4}}}}={\frac{{{\ln{{10}}}}}{{{\ln{\ }}{e}}}}\)
The above logarithm is a natural logarithm with base e.
The logarithm can be evaluated with a calculator as,
\(\displaystyle{\frac{{{3}{x}}}{{{4}}}}={\frac{{{2.302}}}{{{1}}}}\)
\(\displaystyle{3}{x}={4}{\left({2.302}\right)}\)
\(\displaystyle{3}{x}={9.208}\)
Further simplify the above equation \(\displaystyle{3}{x}={9.208}\) as,
\(\displaystyle{3}{x}={9.208}\)
\(\displaystyle{x}={\frac{{{9.208}}}{{{3}}}}\)
\(\displaystyle\approx{3.06}\)
Thus, the solution of the exponential equation \(\displaystyle{e}^{{{\frac{{{3}{x}}}{{{4}}}}}}={10}\) is \(\displaystyle{x}={3.06}\)
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