Question

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

Decimals
Exponential and Logarithmic Equations solve the equation. Find the exact solution if possible. otherwise, use a calculator to approximate to two decimals.
$$\displaystyle{e}^{{{3}\frac{{x}}{{4}}}}={10}$$

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 $$\displaystyle{x}={3.06}$$