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:

$\log _{4}\left( 3x-7\right) ^{2}=10$

The above equation is valid only if $\left( 3x-7\right) ^{2}>0$ or $x\neq \displaystyle \frac{7}{3}.$ The domain is the set of real numbers not equal to $$\frac{7}{ 3}.$$

Convert the equation to an exponential equation with base 4.

$$\begin{eqnarray*}&& \\ \log _{4}\left( 3x-7\right) ^{2} &=&10 \\ && \\ && \\ 4^{10} &=&\left( 3x-7\right) ^{2} \\ && \\ \end{eqnarray*}$$
$$\begin{eqnarray*}&& \\ \pm 4^{5} &=&3x-7 \\ && \\ && \\ 7\pm 4^{5} &=&3x \\ && \\ \end{eqnarray*}$$
$$\begin{eqnarray*}&& \\ \displaystyle \frac{7\pm 4^{5}}{3} &=&\displaystyle \fra... ...&& \\ && \\ x &=&\displaystyle \frac{7\pm 4^{5}}{3} \\ && \\ \end{eqnarray*}$$
$$\begin{eqnarray*}&& \\ x &=&\displaystyle \frac{7+4^{5}}{3}=\displaystyle \frac... ... x &=&\displaystyle \frac{7-4^{5}}{3}=-339 \\ && \\ && \\ && \end{eqnarray*}$$

The exact answers are $x=\displaystyle \frac{1031}{3}$ and $-339.\bigskip\bigskip$

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 $x=\displaystyle \frac{1031}{3}$ by substituting $$\frac{1031}{3}$$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: $\qquad \log _{4}\left( 3x-7\right) ^{2}=\log _{4}\left( 3\left( \displaystyle \frac{1031}{3}\right) -7\right) ^{2}$
  $$\begin{eqnarray*}&& \\ &&\log _{4}\left( \left( 1031\right) -7\right) ^{2} \\ && \\ &=&\log _{4}\left( 1,048,576\right) \\ && \\ \end{eqnarray*}$$
  $$\begin{eqnarray*}&& \\ &\approx &\displaystyle \frac{\ln \left( 1,048,576\right) }{\ln \left( 4\right) }\approx 10 \\ && \end{eqnarray*}$$

  
  

  
  

• Right Side:$\qquad 10$

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value $$\frac{1031}{3}$$ for x, then $x=\displaystyle \frac{1031}{3}$ is a solution.

Check the answer x=-339 by substituting -339 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: $\qquad \log _{4}\left( 3x-7\right) ^{2}=\log _{4}\left( 3\left( -339\right) -7\right) ^{2}$
  $$\begin{eqnarray*}&& \\ &&\log _{4}\left( -1,024\right) ^{2} \\ && \\ &=&\log _{4}\left( 1,048,576\right) \\ && \\ \end{eqnarray*}$$
  $$\begin{eqnarray*}&& \\ &\approx &\displaystyle \frac{\ln \left( 1,048,576\right) }{\ln \left( 4\right) }\approx 10 \\ && \end{eqnarray*}$$

  
  

• Right Side:$\qquad 10$

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value -339 for x, then x=-339 is a solution.

Graphical Check:

You can also check your answer by graphing $\quad f(x)=\log _{4}\left( 3x-7\right) ^{2}-10\quad$(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 343.6666666667 and -339. This means that 343.6666666667 and -339 are the real solutions.

You may have to change the original equation somewhat to graph it because most graphing calculators only have the natural log function and the common log function. Rewrite the original equation $f(x)=\log _{4}\left( 3x-7\right) ^{2}-10$ in the equivalent form $f(x)=\displaystyle \frac{\log \left( 3x-7\right) ^{2}}{\log \left( 4\right) }-10$ and graph it

Recall that $\log _{4}\left( 3x-7\right) ^{2}=10$ is equivalent to $\log _{4}\left( 3x-7\right) =5$ only for those values of $x>\displaystyle \frac{7}{3}.$ Why is that? The domain of $\log _{4}\left( 3x-7\right) ^{2}=10$ is the set of real numbers $\neq \displaystyle \frac{7}{3}$, and the domain of $\log _{4}\left( 3x-7\right) =5$ is the set of real numbers $>\displaystyle \frac{7}{3}$

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