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 functions.

Solve for x in the following equation.

Problem 8.4a:

$\log _{8}\left( 3x+4\right) ^{4}=24$

Answers:

x=87,380 and $x=-\displaystyle \frac{262,148}{3}\bigskip \bigskip$

Solution:

The above equation is valid only if $\quad \log _{8}\left( 3x+4\right) ^{4}$is valid. The term $\log _{8}\left( 3x+4\right) ^{4}$ is valid if $\left( 3x+4\right) ^{4}>0\longrightarrow x\neq -\displaystyle \frac{4}{3}.$ Therefore, the equation is valid when $x\neq -\displaystyle \frac{4}{3}.$ Another way of saying this is that the domain is the set of real numbers where $x\neq -\displaystyle \frac{4}{3}.$

If you choose to work the problem by first removing the exponent 4, you will lose one of the solutions because $\log _{8}\left( 3x+4\right) ^{4}$ is equivalent to $4\log _{8}\left( 3x+4\right)$ only when $x>-\displaystyle \frac{4}{3}$.

Covert the logarithmic equation to an exponential equation.

$$\begin{eqnarray*}&& \\ \log _{8}\left( 3x+4\right) ^{4} &=&24 \\ && \\ && \\ 8^{24} &=&\left( 3x+4\right) ^{4} \\ && \\ \end{eqnarray*}$$
$$\begin{eqnarray*}&& \\ \pm 8^{6} &=&3x+4 \\ && \\ && \\ \pm 8^{6}-4 &=&3x \\ && \\ \end{eqnarray*}$$
$$\begin{eqnarray*}&& \\ \displaystyle \frac{\pm 8^{6}-4}{3} &=&\displaystyle \fr... ...& \\ && \\ x &=&\displaystyle \frac{\pm 8^{6}-4}{3} \\ && \\ \end{eqnarray*}$$
$$\begin{eqnarray*}&& \\ x &=&\displaystyle \frac{8^{6}-4}{3}=87,380 \\ && \\ &... ...ystyle \frac{262,148}{3}\approx -87,382.6666666667 \\ && \\ && \end{eqnarray*}$$

The exact answers are $x=\displaystyle \frac{\pm 8^{6}-4}{3}.\bigskip\bigskip\bigskip$

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

Numerical Check:

• Check the answer $x=\displaystyle \frac{8^{6}-4}{3}$ by substituting 87,380 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:

$\qquad \log _{8}\left( 3x+4\right) ^{4}=\log _{8}\left( 3\left( 87,380\right) +4\right) ^{4}$

$$\begin{eqnarray*}&& \\ &=&\log _{8}\left( 4.72236648287\times 10^{21}\right) \\... ... 10^{21}\right) }{\ln \left( 8\right) } \\ && \\ &=&24 \\ && \end{eqnarray*}$$

Right Side:$\qquad 24$

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value 87,380 for x, then x=87,380 is a solution.

• Check the answer $x=\displaystyle \frac{-8^{6}-4}{3}$ by substituting $-\displaystyle \frac{ 262,148}{3}$ 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: $\qquad \log _{8}\left( 3x+4\right) ^{4}=\log _{8}\left( 3\left( - \displaystyle \frac{262,148}{3}\right) +4\right) ^{4}$

$$\begin{eqnarray*}&& \\ &=&\log _{8}\left( 4.72236648287\times 10^{21}\right) \\... ... 10^{21}\right) }{\ln \left( 8\right) } \\ && \\ &=&24 \\ && \end{eqnarray*}$$

Right Side:$\qquad 24$

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value $-\displaystyle \frac{ 262,148}{3}$for x, then $x=-\displaystyle \frac{262,148}{3}$ is a solution.

Graphical Check:

You can also check your answer by graphing $\quad f(x)=\log _{8}\left( 3x+4\right) ^{4}-24\quad$(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 87,380 and -87,382.66666667. This means that 87,380 and -87,382.66666667 are the real solutions.

If you have trouble graphing the function $f(x)=\log _{8}\left( 3x+4\right) ^{4}-24$, graph the equivalent function $f(x)=\displaystyle \frac{\ln \left( 3x+4\right) ^{4}}{\ln \left( 8\right) }-24$.

If you would like to review the solution to problem 8.4b, click on solution.

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