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 4:

$$\log_{7}\left(\frac{3x}{11}\right)=2$$

Note that the domain of $$\log_{7}\left(\frac{3x}{11}\right)=2$$ is the set of real numbers greater than zero because you cannot take the log of zero or a negative number.

$$\begin{eqnarray*}\log_{7}\left(\frac{3x}{11}\right)&=&2\\ && \\ 7^{2}&=&\frac{3x}{11} \end{eqnarray*}$$

$$\begin{eqnarray*}11\cdot\,49&=&3x\\ &&\\ \frac{11\cdot\,49}{3}&=&x \end{eqnarray*}$$

$$\begin{eqnarray*}x&=&\frac{539}{3}\\ &&\\ x&\approx&179.6666666667 \end{eqnarray*}$$

The exact value is $$\frac{539}{3}$$ and the approximate value is $x\approx 179.6666666667$

Check the solution $$\frac{539}{3}$$ by substituting $x\approx 179.6666666667$ 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: $$\log_{7}\left(\frac{3x}{11}\right)$$
  $$\begin{eqnarray*}&=& \log_{7}\left(\displaystyle {\frac{\displaystyle {\frac{3\c... ...}}}{11}}\right)\\ &&\\ &=& \log_{7}\left(\frac{359}{11}\right) \end{eqnarray*}$$

  
  

  $$\begin{eqnarray*}&=&\displaystyle {\frac{\ln\left(\displaystyle{\frac{359}{11}}\right)}{\ln\,7}}\\ &&\\ &\approx& 2 \end{eqnarray*}$$
  
  

• Right Side: 2

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value $$\frac{539}{3}$$ for x, then $$x=\frac{539}{3}$$ is a solution.

You can also check your answer by graphing $$f(x)=\log_{7}\left(\frac{3x}{11}\right)-2$$ (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 179.6666666667. This means that 179.6666666667is the real solution.

If you have trouble graphing $$f(x)=\log_{7}\left(\frac{3x}{11}\right)-2$$try graphing the equivalent equation $$\displaystyle{\frac{\log\left(\displaystyle{\frac{3x}{11}}\right)}{\log\;7}}-2$$

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 equation table of contents, click on Contents.

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