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# Using Properties of Logarithms

## Using Properties of Logarithms 1.13 - Solution

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.
$4 \cdot \log(x)=\text{-}12$
$\log(x)=\text{-}3$
$10^{\text{LHS}}=10^{\text{RHS}}$
$10^{\log(x)} = 10^{\text{-}3}$
$10^{\log(m)}=m$
$x = 10^{\text{-}3}$
$x = 0.001$
b

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

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