Solve for x in the following equation.

Problem 8.1b:

$\log _{3}\left( 2x\right) =7$

Answer: The exact answer is $x=\displaystyle \frac{3^{7}}{2}$ or $x=1,093.5.\bigskip\bigskip $


Note that the domain of log3(2x) 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 $x=\displaystyle \frac{3^{7}}{2}$ or $x=1,093.5.\bigskip\bigskip
\bigskip $

Check the solution $x=\displaystyle \frac{3^{7}}{2}$ by substituting x=1,093.5 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 1,093.5 for x, then 1,093.5 is a solution.

You can also check your answer by graphing $\quad f(x)=\log _{3}\left(
2x\right) -7\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=1,093.5. This means that x=1,093.5 is the real solution.

If you have trouble graphing the equation $\quad f(x)=\log _{3}\left(
2x\right) -7$, try graphing the equivalent equation $\quad f(x)=\displaystyle \frac{\log
\left( 2x\right) }{\log \left( 3\right) }-7$

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

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