SOLVING LOGARITHMIC EQUATIONS

Note:



Solve for x in the following equation.


Problem 8.1d:

$\log _{2x}\left( 75\right) =3$


Answer: The exact answer is $x=\displaystyle \frac{\sqrt[3]{75}}{2}$ and the approximate answer is $x\approx 2.10858166325.\bigskip
\bigskip $

Solution:


Note that the domain of $\log _{2x}\left( 75\right) $ is the set of positive real numbers not equal to 1 because the base may not be negative and the base may not equal 1.

Convert $\log _{2x}\left( 75\right) =3$ to an exponential equation with base 2x.




The exact value is $x=\displaystyle \frac{\sqrt[3]{75}}{2}$ and the approximate answer is $x\approx 2.10858166325.\bigskip\bigskip\bigskip $




Check the solution $x=\displaystyle \frac{\sqrt[3]{75}}{2}$ by substituting $x\approx
2.10858166325$ 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.

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value 2.10858166325 for x, then 2.10858166325 is a solution.





You can also check your answer by graphing $\quad f(x)=\log _{2x}\left(
75\right) -3\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=2.10858166325. This means that $x\approx 2.108581663253$ is the real solution.



If you have a difficult time graphing the equation $\quad f(x)=\log
_{2x}\left( 75\right) -3,$ try graphing the equivalent equation $\quad f(x)=
\displaystyle \frac{\log \left( 75\right) }{\log \left( 2x\right)
}-3$




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