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 \(\displaystyle{x}={3.06}\)

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 \(\displaystyle{x}={3.06}\)