SOLVING EXPONENTIAL EQUATIONS


Note:

If you would like an in-depth review of exponents, the rules of exponents, exponential functions and exponential equations, click on exponential function.

Solve for x in the following equation.


Problem 7.5d: tex2html_wrap_inline145


Answer: tex2html_wrap_inline155 The exact solution is and the approximate solution is tex2html_wrap_inline149


Solution:

The exponential term is already isolated.

Take the natural logarithm of both sides of the equation tex2html_wrap_inline151


eqnarray42


eqnarray49

The exact answer is tex2html_wrap_inline155 and the approximate answer is tex2html_wrap_inline155

When solving the above problem, you could have used any logarithm. For example, let's solve it using the logarithm with base 6.


eqnarray58


eqnarray69


eqnarray80

Check this answer in the original equation.

Check the solution by substituting -2.98759384204 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 -2.98759384204 for x, then x=-2.98759384204 is a solution.

You can also check your answer by graphing tex2html_wrap_inline173 (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 -2.98759384204. This means that -2.98759384204 is the real solution.


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