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 8.7b:

$15\log _{80}\left( 4x^{2}-7x-10\right) =60$


Answers: The exact answers are The exact answers are $x=\displaystyle \frac{7\pm \sqrt{655,360,209}}{8}.$ The approximate answers are $x\approx 3200.875510$ and -3199.125510


Solution:

The above equation is valid only if $\quad \log _{80}\left( 4x^{2}-7x-10\right) $ is valid. The term $\log _{80}\left( 4x^{2}-7x-10\right) $ is valid if $\left( 4x^{2}-7x-10\right) >0\longrightarrow x>\displaystyle \frac{7+\sqrt{209}}{8}\approx 2.6821$ or $x<\displaystyle \frac{7- \sqrt{209}}{8}\approx -0.9321.$ Therefore, the equation is valid when the domain is the set of real numbers less than $\displaystyle \frac{7-\sqrt{209}}{8}$ or greater than $\displaystyle \frac{7+\sqrt{209}}{8}.$



Isolate the log term.        

$\log _{80}\left( 4x^{2}-7x-10\right) =4$



Convert the equation to an exponential equation with base 80.        

804=4x2-7x-10


Set the equation equal to zero.        

4x2-7x-40,960,010=0.



Solve for x.        

$x=\displaystyle \frac{7\pm \sqrt{\left( -7\right) ^{2}-4\left( 4\right) \left( ... ...\right) }}{2\left( 4\right) }=\displaystyle \frac{7\pm \sqrt{ 655,360,209}}{8} $


The exact answers are $x=\displaystyle \frac{7\pm \sqrt{655,360,209}}{8}.$

The approximate answers are $x\approx 3200.875510$ and -3199.125510


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.


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