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 1:

$6\log \left( x^{2}+1\right) -x=0$

Isolate the log term..

$$\begin{eqnarray*}&& \\ 6\log \left( x^{2}+1\right) -x &=&0 \\ && \\ && \\ \l... ... && \\ x^{2} &=&10^{\displaystyle \frac{x}{6}}-1 \\ && \\ && \end{eqnarray*}$$

For a beginning student, graphing is the best method. When you graph the function $6\log \left( x^{2}+1\right) -x$, you will notice that the graph crosses the x-axis three times: at 0, at 0.4161031784, and at 13.62667385. These are the three real solutions.

The exact answers are $\ x=3\pm \sqrt{25+e^{5}}$ and the approximate answers are 16.16864 and -10.16864.

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=0 by substituting 0 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: $\qquad 6\log \left( x^{2}+1\right) -x=6\log \left( 0+1\right) -0=0$

• 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 0 for x, then x=0 is a solution.

Check the answer x=0.4161031784 by substituting 0.4161031784 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: $\qquad 6\log \left( x^{2}+1\right) -x=6\log \left( \left( 0.4161031784\right) ^{2}+1\right) -0.41610317840=0$

• 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 0.4161031784 for x, then x=0.4161031784 is a solution.

Check the answer x=13.62667385 by substituting 13.62667385 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: $\qquad 6\log \left( x^{2}+1\right) -x=6\log \left( \left( 13.62667385\right) ^{2}+1\right) -13.62667385=0$

• 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 013.62667385 for x, then x=13.62667385 is a solution.

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.

If you would like to go back to the equation table of contents, click on contents.

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