`3^(2x) = 75` Solve the equation accurate to three decimal places

Monday, February 2, 2015

$\displaystyle{{3}}^{{{2}{x}}}={75}$ Solve the equation accurate to three decimal places

Problem: $\displaystyle{{3}}^{{{2}{x}}}={75}$ is an exponential equation.

To simplify, we need to apply logarithm property: $\displaystyle{\log{{\left({{x}}^{{y}}\right)}}}={y}\cdot{\log{{\left({x}\right)}}}$

to bring down the exponent that is in terms of x.

Taking "log" on both sides:

$\displaystyle{\log{{\left({{3}}^{{{2}{x}}}\right)}}}={\log{{\left({75}\right)}}}$

$\displaystyle{\left({2}{x}\right)}{\log{{\left({3}\right)}}}={\log{{\left({75}\right)}}}$

Divide both sides by $\displaystyle{\log{{\left({3}\right)}}}$ to isolate " $\displaystyle{2}{x}$ ":

$\displaystyle\frac{{{2}{x}\cdot{\log{{\left({3}\right)}}}}}{{{\log{{\left({3}\right)}}}}}=\frac{{{\log{{\left({75}\right)}}}}}{{{\log{{\left({3}\right)}}}}}$

$\displaystyle{2}{x}=\frac{{{\log{{\left({75}\right)}}}}}{{{\log{{\left({3}\right)}}}}}$

Multiply both sides by 1/2 to isolate x:

$\displaystyle{\left(\frac{{1}}{{2}}\right)}\cdot{2}{x}=\frac{{{\log{{\left({75}\right)}}}}}{{{\log{{\left({3}\right)}}}}}\cdot{\left(\frac{{1}}{{2}}\right)}$

Note: You will get the same result when you divide both sides by 2.

The equation becomes:

$\displaystyle{x}=\frac{{{\log{{\left({75}\right)}}}}}{{{2}{\log{{\left({3}\right)}}}}}$

$\displaystyle{x}\approx{1.965}$ Rounded off to three decimal places

To check, plug-in $\displaystyle{x}={1.965}$ in $\displaystyle{{3}}^{{{2}{x}}}={75}$ :

$\displaystyle{{3}}^{{{2}\cdot{1.965}}}=?{75}$

$\displaystyle{{3}}^{{{3.93}}}=?{75}$

$\displaystyle{75.0043637}\approx{75}$ TRUE

Conclusion: $\displaystyle{x}\approx{1.965}$ is the final answer.

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Monday, February 2, 2015

$\displaystyle{{3}}^{{{2}{x}}}={75}$ Solve the equation accurate to three decimal places

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Thomas Jefferson's election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?

Monday, February 2, 2015

Solve the equation accurate to three decimal places

No comments:

Post a Comment

Thomas Jefferson&#39;s election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?

$\displaystyle{{3}}^{{{2}{x}}}={75}$ Solve the equation accurate to three decimal places

Thomas Jefferson's election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?