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.

Example 2:

$3\log _{3}\left( x^{2}-7x+2\right) -5=0$

The equation is valid only if x²-7x+2>0 or $\left( x-\displaystyle \frac{7+\sqrt{41} }{2}\right)$ $\left( x-\displaystyle \frac{7-\sqrt{41}}{2}\right) >0\rightarrow$

$$\begin{eqnarray*}&& \\ x &>&\displaystyle \frac{7+\sqrt{41}}{2}\qquad or\qquad x<\displaystyle \frac{7-\sqrt{41}}{2}. \\ && \\ && \end{eqnarray*}$$

Isolate the log term..

$$\begin{eqnarray*}&& \\ 3\log _{3}\left( x^{2}-7x+2\right) -5 &=&0 \\ && \\ &&... ...&& \\ && \\ 3^{\displaystyle \frac{5}{3}} &=&x^{2}-7x+2 \\ && \end{eqnarray*}$$

$$\begin{eqnarray*}&& \\ x^{2}-7x+2-3^{\displaystyle \frac{5}{3}} &=&0 \\ && \\ ... ...7\pm \sqrt{49-8+4\cdot 3^{\displaystyle \frac{5}{3}}}}{2} \\ && \end{eqnarray*}$$

$$\begin{eqnarray*}&& \\ x &=&\displaystyle \frac{7\pm \sqrt{41+4\cdot 3^{\displa... ...laystyle \frac{5}{3}}}}{2}\approx -0.560819 \\ && \\ && \\ && \end{eqnarray*}$$

The exact answers are $x=\displaystyle \frac{7\pm \sqrt{41+4\cdot 3^{\displaystyle \frac{5}{3}}}}{2}$and the approximate answers are 7.560819 and -0.560819.

These answers may or may not be the solutions. You must check them with the original equation, either by a numerical substitution or by graphing.

Numerical Check:

Check the answer $x=\displaystyle \frac{7+\sqrt{41+4\cdot 3^{\displaystyle \frac{5}{3}}}}{2}$ by substituting 7.560819 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: $\qquad 3\log _{3}\left( x^{2}-7x+2\right) -5=3\log _{3}\left( \left( 7.560819\right) ^{2}-7\left( 7.560819\right) +2\right) -5$
  $$\begin{eqnarray*}&& \\ &=&3\log _{3}\left( 6.240251\right) -5 \\ && \\ &=&3\d... ...left( 6.240251\right) }{\log \left( 3\right) }-5 \\ && \\ &=&0 \end{eqnarray*}$$
  
  

• Right Side:$\qquad 0$

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value 7.560819 for x, then x=7.560819 is a solution.

Check the answer $x=\displaystyle \frac{7-\sqrt{41+4\cdot 3^{\displaystyle \frac{5}{3}}}}{2}$ by substituting -0.560819 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: $\qquad 3\log _{3}\left( x^{2}-7x+2\right) -5=3\log _{3}\left( \left( -0.560819\right) ^{2}-7\left( -0.560819\right) +2\right) -5$
  $$\begin{eqnarray*}&& \\ &=&3\log _{3}\left( 6.240251\right) -5 \\ && \\ &=&3\d... ...left( 6.240251\right) }{\log \left( 3\right) }-5 \\ && \\ &=&0 \end{eqnarray*}$$
  
  

• Right Side:$\qquad 0$

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value 7.560819 for x, then x=7.560819 is a solution.

Graphical Method:

Graph the function

$f(x)=3\log _{3}\left( x^{2}-7x+2\right) -5$ or $f(x)=3 \displaystyle \frac{\log \left( x^{2}-7x+2\right) }{\log (3)}-5$ and look to see where the graph crosses the x-axis. Note that the graph crosses at two places and the two places are the two real solutions.

If you would like to work another example, click on example.

If you would like to test yourself by working some problems similar to this example, click on problem.

If you would like to go back to the previous section, click on previous.

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