`log_5sqrt(x - 4) = 3.2` Solve the equation accurate to three decimal places

Tuesday, February 22, 2011

$\displaystyle{\log}_{{5}}\sqrt{{{x}-{4}}}={3.2}$ Solve the equation accurate to three decimal places

To simplify the logarithmic equation: $\displaystyle{\log}_{{5}}{\left(\sqrt{{{x}-{4}}}\right)}={3.2}$ , recall the logarithm property: $\displaystyle{{a}}^{{{\left({\log}_{{{a}}}{\left({x}\right)}\right)}}}={x}$ .

When a logarithm function is raised by the same base, the log cancels out which is what we need to do on the left side of the equation.

As a rule we apply same change on both sides of the equation.

Raising both sides by base of 5:

$\displaystyle{{5}}^{{{\log}_{{5}}{\left(\sqrt{{{x}-{4}}}\right)}}}={{5}}^{{{3.2}}}$

$\displaystyle\sqrt{{{x}-{4}}}={{5}}^{{{3.2}}}$

To cancel out the radical sign, square both sides:

$\displaystyle{{\left(\sqrt{{{x}-{4}}}\right)}}^{{2}}={{\left({{5}}^{{{3.2}}}\right)}}^{{2}}$

$\displaystyle{x}-{4}={{5}}^{{{6.4}}}$

$\displaystyle{x}={{5}}^{{{6.4}}}+{4}$

$\displaystyle{x}\approx{29748.593}$ (rounded off to three decimal places)

To check, plug-in $\displaystyle{x}={29748.593}$ in $\displaystyle{\log}_{{5}}{\left(\sqrt{{{x}-{4}}}\right)}$ :

$\displaystyle{\log}_{{5}}{\left(\sqrt{{{29748.593}-{4}}}\right)}$

$\displaystyle{\log}_{{5}}{\left(\sqrt{{{29744.593}}}\right)}$

$\displaystyle{\log}_{{5}}{\left({172.4662083}\right)}={3.2}$ which is what we want

So, x=29748.593 is the real solution.

Note: $\displaystyle{{\left({{x}}^{{m}}\right)}}^{{n}}={{x}}^{{{\left({m}\cdot{n}\right)}}}$

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Tuesday, February 22, 2011

$\displaystyle{\log}_{{5}}\sqrt{{{x}-{4}}}={3.2}$ 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?

Tuesday, February 22, 2011

Solve the equation accurate to three decimal places

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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?

$\displaystyle{\log}_{{5}}\sqrt{{{x}-{4}}}={3.2}$ 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?