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Solving Logarithmic Equations

In order to solve logarithmic equations, you will take advantage of base-10 logarithms and the powers of ten cancelling each other out. For example, an equation in the form 5lg(x)+2=12 5 \cdot \lg (x) + 2 = 12 can be solve using this method.

Isolate the logarithm with the unknown variable so that it is by itself on the right or left-hand side.
5lg(x)+2=125 \cdot \lg (x) + 2 = 12
5lg(x)=105 \cdot \lg (x) = 10
lg(x)=2\lg (x) = 2
Because the left and right hand side of the equation need to be equal, 1010 raised by the value on the left hand side should be equal to 1010 raised to the right hand side. This is used in order to cancel out the logarithms. So both sides are rewritten in base 1010:

10lg(x)=102. 10^{\lg (x)} = 10^2.

The power of ten "cancels out" the logarithm and leaves only what is outside the logarithm, in other words x.x.
10lg(x)=10210^{\lg (x)} = 10^2
10lg(a)=a10^{\lg(a)}=a
x=102x = 10^2
x=100x = 100