1
Here, we show you a step-by-step solved example of properties of logarithms. This solution was automatically generated by our smart calculator:
log β‘ x β
y β
z 3 \log\sqrt[3]{x\cdot y\cdot z} log 3 x β
y β
z β
2
Using the power rule of logarithms: log β‘ a ( x n ) = n β
log β‘ a ( x ) \log_a(x^n)=n\cdot\log_a(x) log a β ( x n ) = n β
log a β ( x )
1 3 log β‘ ( x y z ) \frac{1}{3}\log \left(xyz\right) 3 1 β log ( x yz )
3
Use the product rule for logarithms: log β‘ b ( M N ) = log β‘ b ( M ) + log β‘ b ( N ) \log_b\left(MN\right)=\log_b\left(M\right)+\log_b\left(N\right) log b β ( MN ) = log b β ( M ) + log b β ( N ) M = x M=x M = x N = y z N=yz N = yz
1 3 ( log β‘ ( x ) + log β‘ ( y z ) ) \frac{1}{3}\left(\log \left(x\right)+\log \left(yz\right)\right) 3 1 β ( log ( x ) + log ( yz ) )
4
Use the product rule for logarithms: log β‘ b ( M N ) = log β‘ b ( M ) + log β‘ b ( N ) \log_b\left(MN\right)=\log_b\left(M\right)+\log_b\left(N\right) log b β ( MN ) = log b β ( M ) + log b β ( N ) M = y M=y M = y N = z N=z N = z
1 3 ( log β‘ ( x ) + log β‘ ( y ) + log β‘ ( z ) ) \frac{1}{3}\left(\log \left(x\right)+\log \left(y\right)+\log \left(z\right)\right) 3 1 β ( log ( x ) + log ( y ) + log ( z ) )
5
Multiply the single term 1 3 \frac{1}{3} 3 1 β ( log β‘ ( x ) + log β‘ ( y ) + log β‘ ( z ) ) \left(\log \left(x\right)+\log \left(y\right)+\log \left(z\right)\right) ( log ( x ) + log ( y ) + log ( z ) )
1 3 log β‘ ( x ) + 1 3 log β‘ ( y ) + 1 3 log β‘ ( z ) \frac{1}{3}\log \left(x\right)+\frac{1}{3}\log \left(y\right)+\frac{1}{3}\log \left(z\right) 3 1 β log ( x ) + 3 1 β log ( y ) + 3 1 β log ( z )
ξ Final answer to the exercise
1 3 log β‘ ( x ) + 1 3 log β‘ ( y ) + 1 3 log β‘ ( z ) \frac{1}{3}\log \left(x\right)+\frac{1}{3}\log \left(y\right)+\frac{1}{3}\log \left(z\right) 3 1 β log ( x ) + 3 1 β log ( y ) + 3 1 β log ( z ) ξ