`2^(3-z) = 625` Solve the equation accurate to three decimal places

Monday, January 26, 2015

$\displaystyle{{2}}^{{{3}-{z}}}={625}$ Solve the equation accurate to three decimal places

For exponential equation: $\displaystyle{{2}}^{{{3}-{z}}}={625}$ , we may apply the logarithm property:

$\displaystyle{\log{{\left({{x}}^{{y}}\right)}}}={y}\cdot{\log{{\left({x}\right)}}}$ .

This helps to bring down the exponent value.

Taking "log" on both sides:

$\displaystyle{\log{{\left({{2}}^{{{3}-{z}}}\right)}}}={\log{{\left({625}\right)}}}$

$\displaystyle{\left({3}-{z}\right)}\cdot{\log{{\left({2}\right)}}}={\log{{\left({625}\right)}}}$

Divide both sides by log (2) to isolate (3-z):

$\displaystyle\frac{{{\left({3}-{z}\right)}\cdot{\log{{\left({2}\right)}}}}}{{{\log{{\left({2}\right)}}}}}=\frac{{{\log{{\left({625}\right)}}}}}{{{\log{{\left({2}\right)}}}}}$

$\displaystyle{3}-{z}=\frac{{{\log{{\left({625}\right)}}}}}{{{\log{{\left({2}\right)}}}}}$

Subtract both sides by 3 to isolate "-z":

$\displaystyle{3}-{z}=\frac{{{\log{{\left({625}\right)}}}}}{{{\log{{\left({2}\right)}}}}}$

-3 -3

------------------------------------

$\displaystyle-{z}=\frac{{{\log{{\left({625}\right)}}}}}{{{\log{{\left({2}\right)}}}}}-{3}$

Multiply both sides by -1 to solve +z or z:

$\displaystyle{\left(-{1}\right)}\cdot{\left(-{z}\right)}={\left(-{1}\right)}\cdot{\left[\frac{{{\log{{\left({625}\right)}}}}}{{{\log{{\left({2}\right)}}}}}-{3}\right]}$

$\displaystyle{z}\approx-{6.288}$ Rounded off to three decimal places.

To check, plug-in $\displaystyle{z}=-{6.288}$ in $\displaystyle{{2}}^{{{3}-{z}}}={625}$ :

$\displaystyle{{2}}^{{{3}-{\left(-{6.288}\right)}}}=?{625}$

$\displaystyle{{2}}^{{{3}+{6.288}}}=?{625}$

$\displaystyle{{2}}^{{{9.288}}}=?{625}$

$\displaystyle{625.1246145}\approx{625}$ TRUE

Conclusion: $\displaystyle{z}\approx-{6.288}$ as the final answer.

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Monday, January 26, 2015

$\displaystyle{{2}}^{{{3}-{z}}}={625}$ 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, January 26, 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?

$\displaystyle{{2}}^{{{3}-{z}}}={625}$ 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?