 Solving Logarithmic Equations - Maple Help

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 ${log}_{b}\left(x\right)=y$ (that is, involving just a single logarithm) then $x={b}^{y}$.

Example: Solve ${\mathrm{log}}_{4}\left(x\right)\cdot {\mathrm{log}}_{2}\left({x}^{2}\right)=16$

Solution: First, note that by the power rule , so the original equation reduces to ${\mathrm{log}}_{4}\left(x\right)\cdot {\mathrm{log}}_{2}\left(x\right)=8$. Next, using the change of base rule, we have ${\mathrm{log}}_{4}\left(x\right)=\frac{{\mathrm{log}}_{2}\left(x\right)}{{\mathrm{log}}_{2}\left(4\right)}=\frac{{\mathrm{log}}_{2}\left(x\right)}{2}$. Substituting this into ${\mathrm{log}}_{4}\left(x\right)\cdot {\mathrm{log}}_{2}\left(x\right)=8$ and cross-multiplying by 2, we get ${\mathrm{log}}_{2}\left(x\right)\cdot {\mathrm{log}}_{2}\left(x\right)={\left({\mathrm{log}}_{2}\left(x\right)\right)}^{2}=16$. Taking the square roots of both sides gives ${\mathrm{log}}_{2}\left(x\right)=±4$. So, there are two solutions: $x={2}^{4}=16$ and $x={2}^{-4}=\frac{1}{16}$.

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 $y={\mathrm{log}}_{3}\left({x}^{2}\right)$ is $x\ne 0$, while the domain of  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.

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