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:

$\log _{6}\left( 4x+5\right) ^{3}=6$

The above equation is valid only if $\left( 4x+5\right) ^{3}>0$ or $x>- \displaystyle \frac{5}{4}.$ The domain is the set of real numbers greater than $-\displaystyle \frac{5}{ 4}.$

Simplify the left side of the equation using the rules of logarithms.

$$\begin{eqnarray*}&& \\ \log _{6}\left( 4x+5\right) ^{3} &=&6 \\ && \\ && \\ 3\log _{6}\left( 4x+5\right) &=&6 \\ && \\ \end{eqnarray*}$$
$$\begin{eqnarray*}&& \\ \displaystyle \frac{3\log _{6}\left( 4x+5\right) }{3} &=... ...\\ && \\ \log _{6}\left( 4x+5\right) &=&2 \\ && \\ && \\ && \end{eqnarray*}$$

Convert the equation to an exponential equation with base 6.

$$\begin{eqnarray*}&& \\ \log _{6}\left( 4x+5\right) &=&2 \\ && \\ && \\ 6^{2} &=&4x+5 \\ && \\ \end{eqnarray*}$$
$$\begin{eqnarray*}&& \\ 36 &=&4x+5 \\ && \\ && \\ 31 &=&4x \\ && \\ \end{eqnarray*}$$
$$\begin{eqnarray*}&& \\ x &=&\displaystyle \frac{31}{4}=7.75 \\ && \\ && \\ && \end{eqnarray*}$$

The exact answer is $x=\displaystyle \frac{31}{4}=7.75.\bigskip\bigskip\bigskip$

This answer may or may not be the solution to the original equation. You must check this answer by substitution or by graphing with the original equation.

Numerical Check:

Check the answer x=7.75 by substituting 7.75 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: $\qquad \log _{6}\left( 4x+5\right) ^{3}=\log _{6}\left( 4\left( 7.75\right) +5\right) ^{3}=\log _{6}\left( 46,656\right)$
  $$\begin{eqnarray*}&& \\ &=&\displaystyle \frac{\ln \left( 46,656\right) }{\ln \left( 6\right) } \\ && \\ &=&6 \\ && \end{eqnarray*}$$

  
  

  
  

• Right Side:$\qquad 6$

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value 7.75 for x, then x=7.75 is a solution.

Graphical Check:

You can also check your answer by graphing $\quad f(x)=\log _{6}\left( 4x+5\right) ^{3}-6\quad$(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 7.75. This means that 7.75 is the real solution.

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 $f(x)=\log _{6}\left( 4x+5\right) ^{3}-6$ in the equivalent form $f(x)=\displaystyle \frac{\log \left( 4x+5\right) ^{3}}{\log \left( 6\right) }-6$ and graph it

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