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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 $5 \cdot \lg (x) + 2 = 12$ 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.
$5 \cdot \lg (x) + 2 = 12$
$5 \cdot \lg (x) = 10$
$\lg (x) = 2$
Because the left and right hand side of the equation need to be equal, $10$ raised by the value on the left hand side should be equal to $10$ raised to the right hand side. This is used in order to cancel out the logarithms. So both sides are rewritten in base $10$:

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

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