Using Properties of Logarithms

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

DivEqn

\left.\text{LHS}\middle/4\right.=\left.\text{RHS}\middle/4\right.

\log(x)=\text{-}3

10^{\text{LHS}}=10^{\text{RHS}}

10^{\log(x)} = 10^{\text{-}3}

10^{\log(m)}=m

x = 10^{\text{-}3}

CalcPowCalculate power

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}

CalcPowCalculate power

4x = 1000

DivEqn

\left.\text{LHS}\middle/4\right.=\left.\text{RHS}\middle/4\right.

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

AddEqn

\text{LHS}+1=\text{RHS}+1

5\log(2x)=10

DivEqn

\left.\text{LHS}\middle/5\right.=\left.\text{RHS}\middle/5\right.

\log(2x)=2

10^{\text{LHS}}=10^{\text{RHS}}

10^{\log(2x)} = 10^{2}

10^{\log(m)}=m

2x = 10^{2}

CalcPowCalculate power

2x = 100

DivEqn

\left.\text{LHS}\middle/2\right.=\left.\text{RHS}\middle/2\right.

x=50

Using Properties of Logarithms 1.13 - Solution