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 7.6b:tex2html_wrap_inline155tex2html_wrap_inline154

Answer:

Exact answer: tex2html_wrap_inline155 tex2html_wrap_inline156 tex2html_wrap_inline155 Approximate answer: tex2html_wrap_inline155 tex2html_wrap_inline158


Solution:

The first step is to isolate tex2html_wrap_inline160


Add 10 to both sides of the equation.


eqnarray36



Divide both sides of the equation by 6.


eqnarray42



The next is to isolate the variable x.


Take the natural logarithm of both sides of the equation tex2html_wrap_inline166


eqnarray53


eqnarray56


eqnarray61


eqnarray71



The exact answer is tex2html_wrap_inline168 and the approximate answer is tex2html_wrap_inline170



When solving the above problem, you could have used any logarithm. For example, let's solve it using the logarithm with base 12.


eqnarray53


eqnarray83


eqnarray91


eqnarray99



Check this answer in the original equation.



Check the solution tex2html_wrap_inline168 by substituting 6.72105705435 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 6.72105705435 for x, then x=6.72105705435 is a solution.


You can also check your answer by graphing tex2html_wrap_inline188 (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 6.72105705435. This means that 6.72105705435 is the real solution.








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