SOLVING EXPONENTIAL EQUATIONS

Note:

• To solve an exponential equation, isolate the exponential term, take the logarithm of both sides and solve.

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.

Problem 7.4a:

Answer:The exact answer are and The approximate answers are and

Solution:

In order to solve this equation, we have to isolate the exponential term. Since we cannot easily do this in the equation's present form, let's tinker with the equation until we have it in a form we can solve.

Factor the left side of the equation

The only way that a product can equal zero is if at least one of the factors is zero.

Now we have an equation where the exponential term is isolated. Take the natural logarithm of both sides of the equation

Now let's look at the second factor,

Now we have a second equation where the exponential term is isolated. Take the natural logarithm of both sides of the equation

The exact answers are and The approximate answers are and

Check these answers in the original equation.

Check the solution by substituting 0.693147180567 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.

• Left Side:

• Right Side:

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value 0.69314718056 for x, then x=0.69314718056 is a solution.

Check the solution by substituting 1.60943791243 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.

Left Side:

Right Side:

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value 1.60943791243 for x, then x=1.60943791243 is a solution.

You can also check your answer by graphing (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 two places: 0.693147180567 and 1.60943791243. This means that 0.693147180567 and 1.60943791243 are the real solutions.

If you would like to review the solution to problem 7.4b, click on solution.

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