SOLVING LOGARITHMIC EQUATIONS

Note:




Solve for x in the following equation.


Problem 8.1a:

$\ln \left( 4x\right) =14$



Answer: $x=\displaystyle \frac{e^{14}}{4},$ and the approximate answer is $x\approx 300,651.071041.\bigskip\bigskip $


Solution:


Note that the domain of $\ln (4x)$ is the set of real numbers greater than zero because you cannot take the log of zero or a negative number.




The exact value is $x=\displaystyle \frac{e^{14}}{4}$ and the approximate value




Check the solution $x=\displaystyle \frac{e^{14}}{4}$ by substituting $x\approx
300,651.071010$ 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.

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value 300,651.071010 for x, then $x\approx
300,651.071010$ is a solution.





You can also check your answer by graphing $\quad f(x)=\ln \left( 4x\right)
-14\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 $x\approx
300,651.071010$. This means that $x\approx
300,651.071010$ is the real solution.



If you would like to review the solution to problem 8.1b, click on Solution.


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