New Notes Blog: `log_4(128)` Evaluate the logarithm. |

Monday, 22 June 2015

$\displaystyle{\log}_{{4}}{\left({128}\right)}$ Evaluate the logarithm. |

$\displaystyle{\log}_{{4}}{\left({128}\right)}$

To evaluate, factor 128.

$\displaystyle={\log}_{{4}}{\left({{2}}^{{7}}\right)}$

Then, apply the formula of change base $\displaystyle{\log}_{{b}}{\left({a}\right)}=\frac{{{\log}_{{c}}{\left({a}\right)}}}{{{\log}_{{c}}{\left({b}\right)}}}$ .

$\displaystyle=\frac{{{\log}_{{2}}{\left({{2}}^{{7}}\right)}}}{{{\log}_{{2}}{\left({4}\right)}}}$

$\displaystyle=\frac{{{\log}_{{2}}{\left({{2}}^{{7}}\right)}}}{{{\log}_{{2}}{\left({{2}}^{{2}}\right)}}}$

To simplify it further, apply the rule $\displaystyle{\log}_{{b}}{\left({{a}}^{{m}}\right)}={m}\cdot{\log}_{{b}}{\left({a}\right)}$ .

$\displaystyle=\frac{{{7}\cdot{\log}_{{2}}{\left({2}\right)}}}{{{2}\cdot{\log}_{{2}}{\left({2}\right)}}}$

When the base and argument of logarithm are the same, it simplifies to 1, $\displaystyle{\log}_{{b}}{\left({b}\right)}={1}$ .

$\displaystyle=\frac{{{7}\cdot{1}}}{{{2}\cdot{1}}}$

$\displaystyle=\frac{{7}}{{2}}$

Therefore, $\displaystyle{\log}_{{4}}{\left({128}\right)}=\frac{{7}}{{2}}$ .

$\displaystyle{\log}_{{4}}{\left({128}\right)}$

To evaluate, factor 128.

$\displaystyle={\log}_{{4}}{\left({{2}}^{{7}}\right)}$

Then, apply the formula of change base $\displaystyle{\log}_{{b}}{\left({a}\right)}=\frac{{{\log}_{{c}}{\left({a}\right)}}}{{{\log}_{{c}}{\left({b}\right)}}}$ .

$\displaystyle=\frac{{{\log}_{{2}}{\left({{2}}^{{7}}\right)}}}{{{\log}_{{2}}{\left({4}\right)}}}$

$\displaystyle=\frac{{{\log}_{{2}}{\left({{2}}^{{7}}\right)}}}{{{\log}_{{2}}{\left({{2}}^{{2}}\right)}}}$

To simplify it further, apply the rule $\displaystyle{\log}_{{b}}{\left({{a}}^{{m}}\right)}={m}\cdot{\log}_{{b}}{\left({a}\right)}$ .

$\displaystyle=\frac{{{7}\cdot{\log}_{{2}}{\left({2}\right)}}}{{{2}\cdot{\log}_{{2}}{\left({2}\right)}}}$

When the base and argument of logarithm are the same, it simplifies to 1, $\displaystyle{\log}_{{b}}{\left({b}\right)}={1}$ .

$\displaystyle=\frac{{{7}\cdot{1}}}{{{2}\cdot{1}}}$

$\displaystyle=\frac{{7}}{{2}}$

Therefore, $\displaystyle{\log}_{{4}}{\left({128}\right)}=\frac{{7}}{{2}}$ .

New Notes Blog

Monday, 22 June 2015

$\displaystyle{\log}_{{4}}{\left({128}\right)}$ Evaluate the logarithm. |

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In "By the Waters of Babylon," under the leadership of John, what do you think the Hill People will do with their society?

Monday, 22 June 2015

Evaluate the logarithm. |

No comments:

Post a Comment

In &quot;By the Waters of Babylon,&quot; under the leadership of John, what do you think the Hill People will do with their society?

$\displaystyle{\log}_{{4}}{\left({128}\right)}$ Evaluate the logarithm. |

In "By the Waters of Babylon," under the leadership of John, what do you think the Hill People will do with their society?