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Using Properties of Logarithms

Using Properties of Logarithms 1.13 - Solution

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a
We begin by isolating the logarithm on one side. When this is done, we will use the inverse properties of logarithms to undo the logarithm.
4log(x)=-124 \cdot \log(x)=\text{-}12
log(x)=-3\log(x)=\text{-}3
10LHS=10RHS10^{\text{LHS}}=10^{\text{RHS}}
10log(x)=10-310^{\log(x)} = 10^{\text{-}3}
10log(m)=m10^{\log(m)}=m
x=10-3x = 10^{\text{-}3}
x=0.001x = 0.001
b

Here, the logarithm already stands by itself on the left-hand side. It does not matter that it is the logarithm of 4x,4x, we proceed in the same way as before. At the end, we solve for x.x.

log(4x)=3\log(4x)=3
10LHS=10RHS10^{\text{LHS}}=10^{\text{RHS}}
10log(4x)=10310^{\log(4x)} = 10^{3}
10log(m)=m10^{\log(m)}=m
4x=1034x = 10^{3}
4x=10004x = 1000
x=250x=250
c
Here, we also need to get the logarithm by itself before we use that we can use a power to undo a logarithm.
5log(2x)1=95\log(2x)-1=9
5log(2x)=105\log(2x)=10
log(2x)=2\log(2x)=2
10LHS=10RHS10^{\text{LHS}}=10^{\text{RHS}}
10log(2x)=10210^{\log(2x)} = 10^{2}
10log(m)=m10^{\log(m)}=m
2x=1022x = 10^{2}
2x=1002x = 100
x=50x=50