e Following Equation.

8/13/2019 e Following Equation.

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Solve for x in the following equation.

Example 1:

The exponential term is already isolated.

Take the natural logarithm of both sides of the equation

The exact answer is and the approximate answer is

When solving the above problem, you could have used any logarithm. For example,

let's solve it using the logarithm with base 5.


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Check this answer in the original equation.

Check the solution by substituting 4.27333311902 in the originalequation 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 4.27666611902 for x, then x=4.27666611902 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 4.27666611902. This means that 4.27666611902 is the real solution.


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Example: 2

Note:

To solve an exponential equation, isolate the exponential term, take thelogarithm 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 onexponential function.under Algebra.


Example 1:

Isolate the exponential term.

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let's solve it using the logarithmic with base 5.


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Check the solution by substituting 4.00733318523 in the original

equation for x. If the left side of the equation equals the right side of the equation after



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Left Side:

Right Side:



a solution.






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Example: 3

Note:


If you would like an in-depth review of exponents, the rules of exponents, exponential functions

and exponential equations, click onexponential function.under Algebra.


Example 1:

Isolate the exponential term.

Divide both sides of the equation by 4



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The exact answer is ( which can also be written ) and the approximate

answer is


let's solve it using the logarithmic with base 5.


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Check the solution

(can also be written in the equivalent form )

by substituting 0.111571775657 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.


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Left Side:

Right Side:


equation after we substitute the value 0.111571775657 for x, then x=0.111571775657

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 0.111571775657. This means that 0.111571775657 is the real

solution.


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Exmaple: 4

Example 1:

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


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Now let's look at the second factor,

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


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The exact answers are x=0 and

Check these answers in the original equation.

Check the solution x=0 by substituting 0 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:


equation after we substitute the value 0 for x, then x=0 is a solution.

Check the solution by substituting 0.69314718056 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.


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Left Side:

Right Side:



a solution.




crosses the x-axis at two places: 0 and 0.69314718056. This means that 0 and

0.69314718056 are the real solutions.


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Example: 5

Note:


If you would like an in-depth review of exponents, the rules of exponents, exponential

functions and exponential equations, click onexponential functionunder Algebra.


Example 1:

The exponential term is already isolated.






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Check the solution by substituting 3.32192809489 in the original




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Left Side: Right Side:


equation after we substitute the value 3.32192809489 for x, then x=3.32192809489 isa solution.






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Example 1:

The first objective is to isolate the expression

Subtract 6 from both sides of the equation.

Divide both sides of the equation by 100.


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The exact answer is and the approximate answer

is




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Check the solution by substituting -0.609853334512 in the original


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Left Side:

Right Side:


equation after we substitute the value -0.609853334512 for x, then x= -

0.609853334512 is a solution.




crosses the x-axis at -0.609853334512. This means that -0.609853334512 is the realsolution.


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Example 1:

The first objective is to isolate the expression

Subtract 6 from both sides of the equation.

Divide both sides of the equation by 100.


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The exact answer is and the approximate answer

is




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Check the solution by substituting -0.609853334512 in the original


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Left Side:

Right Side:


equation after we substitute the value -0.609853334512 for x, then x= -

0.609853334512 is a solution.




crosses the x-axis at -0.609853334512. This means that -0.609853334512 is the realsolution.

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