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. under Algebra.

Solve for x in the following equation.

Example 4:tex2html_wrap_inline155 tex2html_wrap_inline238

Isolate the exponential term.

Note that the base is e, not tex2html_wrap_inline240 Multiply both sides of the equation by tex2html_wrap_inline242


eqnarray37


eqnarray44



Take the natural logarithm of both sides of the equation tex2html_wrap_inline244


eqnarray63


eqnarray68


eqnarray75


eqnarray77


eqnarray83



The exact answer is tex2html_wrap_inline246 and the approximate answer is tex2html_wrap_inline248



Your exact answer may differ dependent how what logarithm you used to solve the problem. However, all forms of the correct answer will simplify to the same approximate answer.


When solving the above problem, you could have used any logarithm. For example, let's solve it using the logarithmic with base 14. Take the tex2html_wrap_inline254 of both sides of the equation tex2html_wrap_inline256


eqnarray63


eqnarray105


eqnarray116


eqnarray122


eqnarray129


eqnarray136


eqnarray143


eqnarray151


eqnarray157



The exact answer is tex2html_wrap_inline246 and the approximate answer is tex2html_wrap_inline248



Check these answers in the original equation.



Check the solution tex2html_wrap_inline246 by substituting 1.11693268486 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 1.11693268486 for x, then x=1.11693268486 is a solution.


You can also check your answer by graphing tex2html_wrap_inline276 (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 1.11693268486.. This means that 1.11693268486 is the real solution.








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