SOLVING LOGARITHMIC EQUATIONS

Note:

If you would like an in-depth review of logarithms, the rules of logarithms, logarithmic functions and logarithmic equations, click on logarithmic function.

Solve for x in the following equation.

Problem 8.1c: $\log _{\displaystyle \frac{2}{3}}\left( -2x\right) =13$

Answer: $x=-\displaystyle \frac{1}{2}\left( \displaystyle \frac{2}{3 }\right) ^{13},$ and the approximate answer is $x\approx -0.002569115543.\bigskip\bigskip$

Solution:

Note that the domain of $\log _{\displaystyle \frac{2}{3}}\left( -2x\right)$ is the set of real numbers such that -2x>0 or $x<0.\bigskip\bigskip$

Convert $\log _{\displaystyle \frac{2}{3}}\left( -2x\right) =13$ to an exponential equation with base $\displaystyle \frac{2}{3}.$

The exact value is $x=-\displaystyle \frac{1}{2}\cdot \left( \displaystyle \frac{2}{3}\right) ^{13}$ and the approximate answer is $x\approx -0.002569115543.\bigskip\bigskip \bigskip$

Check the solution $x=-\displaystyle \frac{1}{2}\cdot \left( \displaystyle \frac{2}{3}\right) ^{13}$ by substituting $x\approx -0.002569115543$ in the original equation for x. If the left side of the equation equals the right side of the equation after the substitution, you have found the correct answer.

Left Side: $\qquad \log _{\displaystyle \frac{2}{3}}\left( -2x\right)$
Right Side: $\qquad 13$

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value -0.002569115543 for x, then -0.002569115543 is a solution.

You can also check your answer by graphing $\quad f(x)=\log _{\displaystyle \frac{2}{3} }\left( -2x\right) -13\quad$ (formed by subtracting the right side of the original equation from the left side). Look to see where the graph crosses the x-axis; that will be the real solution. Note that the graph crosses the x-axis at x=-0.002569115543. This means that $x\approx -0.002569115543$ is the real solution.

If you have trouble graphing the equation $\quad f(x)=\log _{\displaystyle \frac{2}{3} }\left( -2x\right) -13$ , try graphing the equivalent equation $\quad f(x)= \displaystyle \frac{\log \left( -2x\right) }{\log \left( \displaystyle \frac{2}{3}\right) }-13$

If you would like to review the solution to problem 8.1d, click on Solution.

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