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Solving Logarithmic Equations

Basic Technique for Solving Logarithms

If an equation with logarithms can be solved using algebraic techniques, then those techniques will generally involve the product, quotient, and power rules of logarithms—applied in either direction—as well as examining the problem for common bases. If the equation can be manipulated into the form logbx=y (that is, involving just a single logarithm) then x=by.


Example: Solve log4xlog2x2=16

Solution: First, note that by the power rule log2x2=2 log2x, so the original equation reduces to log4xlog2x=8. Next, using the change of base rule, we have log4x=log2xlog24=log2x2. Substituting this into log4xlog2x=8 and cross-multiplying by 2, we get log2xlog2x=log2x2=16. Taking the square roots of both sides gives log2x=±4. So, there are two solutions: x=24=16 and x=24=116.

Caution: When solving equations involving logarithms, it is very important to keep in mind that the domain of a logarithm function is the positive numbers. As we will see in the examples below, algebraic manipulations of expressions involving logarithms can easily lead to "solutions" which are not valid because of this domain restriction. As a simple illustration, observe that the domain of the function y=log3x2 is x0, while the domain of y=2 log3x is x>0. The product rule for logarithms requires that all the logarithms appearing in the rule be properly defined.

Step by step Logarithmic Equation solver

Solve a logarithmic equation interactively using this step-by-step example.


Solving Logarithmic Equations


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