simplify log √ 27 + log 64 log √ 1000/ log 480

asked by guest
on Nov 22, 2024 at 8:37 pm



You asked:

Simplify the expression: log⁡(27)+log⁡(64)log⁡(1000)log⁡(480)\log\left( \sqrt{27} \right) + \frac{\log\left( 64 \right) \log\left( \sqrt{1000} \right)}{\log\left( 480 \right)}

MathBot Answer:

Simplified:

log⁡(27)+log⁡(64)log⁡(1000)log⁡(480)=log⁡(33)+log⁡(64)log⁡(1010)log⁡(480) \log\left( \sqrt{27} \right) + \frac{\log\left( 64 \right) \log\left( \sqrt{1000} \right)}{\log\left( 480 \right)} = \log{\left(3 \sqrt{3} \right)} + \frac{\log{\left(64 \right)} \log{\left(10 \sqrt{10} \right)}}{\log{\left(480 \right)}}

log⁡(27)+log⁡(64)log⁡(1000)log⁡(480)=log⁡(64)log⁡(1000)log⁡(480)+log⁡(33)=log⁡(64)log⁡(1010)log⁡(480)+log⁡(33)=log⁡(33)+9log⁡(2)log⁡(10)log⁡(480)=log⁡(480)log⁡(33)log⁡(480)+9log⁡(2)log⁡(10)log⁡(480)=log⁡((10log⁡(64)480log⁡(3))32log⁡(480))\begin{aligned}\log{\left(\sqrt{27} \right)} + \frac{\log{\left(64 \right)} \log{\left(\sqrt{1000} \right)}}{\log{\left(480 \right)}}&=\frac{\log{\left(64 \right)} \log{\left(\sqrt{1000} \right)}}{\log{\left(480 \right)}} + \log{\left(3 \sqrt{3} \right)}\\&=\frac{\log{\left(64 \right)} \log{\left(10 \sqrt{10} \right)}}{\log{\left(480 \right)}} + \log{\left(3 \sqrt{3} \right)}\\&=\log{\left(3 \sqrt{3} \right)} + \frac{9 \log{\left(2 \right)} \log{\left(10 \right)}}{\log{\left(480 \right)}}\\&=\frac{\log{\left(480 \right)} \log{\left(3 \sqrt{3} \right)}}{\log{\left(480 \right)}} + \frac{9 \log{\left(2 \right)} \log{\left(10 \right)}}{\log{\left(480 \right)}}\\&=\log{\left(\left(10^{\log{\left(64 \right)}} 480^{\log{\left(3 \right)}}\right)^{\frac{3}{2 \log{\left(480 \right)}}} \right)}\end{aligned}

Factored:

log⁡(27)+log⁡(64)log⁡(1000)log⁡(480)=3(log⁡(3)log⁡(480)+6log⁡(2)log⁡(10))2log⁡(480) \log\left( \sqrt{27} \right) + \frac{\log\left( 64 \right) \log\left( \sqrt{1000} \right)}{\log\left( 480 \right)} = \frac{3 \left(\log{\left(3 \right)} \log{\left(480 \right)} + 6 \log{\left(2 \right)} \log{\left(10 \right)}\right)}{2 \log{\left(480 \right)}}


+ Submit Question