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 8.1d:

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

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.

• Left Side: $\qquad \log _{2x}\left( 75\right) \approx \log _{2\left( 2.10858166325\right) }... ...\ln \left( _{2\left( 2.10858166325\right) }\right) }\approx 3\bigskip \bigskip$

  
  

• Right Side:$\qquad 3$

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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