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Related Lessons: Solve Exponential Equations with Logarithms Solving Logarithmic Equations

 

Solving Exponential Equations without Logarithms

 

An exponential equation involves an unknown variable in the exponent. In this lesson, we will focus on the exponential equations that do not require the use of logarithm. In algebra, this topic is also known as solving exponential equations with the same base. Why? The reason is that we can solve the equation by forcing both sides of the exponential equation to have the same or equal base.

There are eight (8) worked examples in this lesson.

 

 

Key Steps

Make the base in both sides of the equation the SAME

so that if bM = bN

then M = N

  • In other words, if you can express the exponential equations to have the same base in both sides then it's OKAY to set their powers or exponents equal to each other.

 

You should also remember the properties of exponents in order to be successful in solving exponential equations. Here they are...

 

Basic Properties of Exponents

1. zero exponent rule of exponents: b^0=1   Zero Property
2. negative exponent rule of exponents: b^-n=1/b^n or 1/b^-n=b^n   Negative Exponent Property
3. product rule of exponents: (b^m)(b^n)=b^m+n   Product Rule
4. quotient rule of exponents: (b^m)/(b^n)=b^(m-n)   Quotient Rule
5. power to a power rule of exponents: (b^m)^n=b^(m*n)   Power to a Power Rule

 

Direction: Solve each exponential equation using the Basic Properties of Exponents.
1. example 1: 5^(3x) = 1/125 See solution
2. example 2: (2^7x)(8)=256 See solution
3. example 3: (64^2x)/(16^2x+3)=1 See solution
4. example 4: (1/36)^(3-x)*(1/6)^x=216 See solution
5. example5: 9^(-3x+2)*3^-x=27^(-2x-1) See solution
6. example 6: 32^(x^2+1)*8^(x-5)=16^2x See solution
7. example 7: (1/49)^-6x*49=√(7)          See solution
8. example 8:[(125)^(-4x+5)]/[(25)^(x^2-5x+1)]=5 See solution

Example 1: Solve the exponential equation 5^(3x) = 1/125.

Solution:

given problem:5^(3x) = 1/125   Given
5^3x =1/5^3  

Express the denominator of the right side with a base of 5. We have 125 = 53.

Apply the Negative Exponent Property.

5^3x=5^-3  

At this point, the bases are the same therefore set the powers equal to each other.

3x=-3   This is just a simple one-step linear equation.
x=-1   To solve x, divide both sides by 3. That's it.

The final answer here is x = −1.

 


 

Example 2: Solve the exponential equation (2^7x)(8)=256 .

Solution:

(2^7x)(8)=256   Given
2^(7x+3)*2^3=(2^8)^x  

Express all numbers with the base of 2. So we have: 8 = 23 and 256 = 28.

Apply the Product Rule on the left, while using the Power to a Power Rule on the right side.

2^(7x+3)=2^8x  

Here we are ready to set the powers equal to each other since we are able to create single bases that are the same on both sides.

7x+3=8x   Solve the simple linear equation.
x=3  

Subtract both sides by 7x to isolate x. Done!

The final answer is x = 3.


 

Example 3: Solve the exponential equation (64^2x)/(16^2x+3)=1.

Solution:

(64^2x)/(16^2x+3)=1   Given
[(2^6)^2x]/[(2^4)^(2x+3)]]=1  

Express each number with a base of 2. In doing so...

64 = 26 and 16 = 24

.

[2^12x/2^(8x+12]=1  

Applying Power to a Power Rule.

In other words, multiply the inner exponent to the outer exponent. Do it for both the numerator and denominator.

2^[12-(8x+12)]=1  

Apply the Quotient Rule.

Subtract the top exponent by the bottom exponent.

2^(4x-12)=1  

This is how it looks after subtracting the exponents.

Now, looking at the right side, can we express 1 as an exponential number with base 2?

The answer is yes! 1 = 20 using the Zero Property of Exponent.

2^(4x-12)=2^0   Now we have the set-up that we want - having the same bases in both sides.
4x-12=0   Set the powers equal to each other, then solve the equation.
x=3  

To solve the equation, start by adding both sides by 12 to move the constant to the right side.

Finally, divide both sides by 4 to get the value of x.

 

Okay, so we find the answer to be x = 3.


 

Example 4: Solve the exponential equation 1/36)^(3-x)*(1/6)^x=216.

Solution:

1/36)^(3-x)*(1/6)^x=216   Given
[1/6^2]^(3-x)*[1/6^1]^x=6^3  

Express each number as an exponential number with base 6.

36 = 62

6 = 61

216 = 63

 

[[6^-2)^(3-x)*(6^-1)^x]=6^2  

Apply the Negative Exponent Property on the left side of the equation.

6^(-6+2x)*6^-x=6^3   Multiply the inner exponents to outer exponents using the Power to a Power Rule.
6^[(-6+2x)+(-x)]=6^3   Since they have a common base, add the exponents using the Product Rule.
6^(-6+x)=6^3   It's obvious that by having a single and equal bases on both sides, we can now set each powers equal to each other.
x=9   Solve the linear equation by adding both sides by 6 to get x = 9.

And so the solution is x = 9.

 


 

Example 5: Solve the exponential equation 9^(-3x+2)*3^-x=27^(-2x-1).

Solution:

9^(-3x+2)*3^-x=27^(-2x-1)   Given
[3^2]^(-3x+2)*3^-x=[3^3]^(-2x-1)  

Use 3 as the common base.

9 = 32 and 27 = 33

Multiply the inner and outer exponents by applying the Power to a Power Rule.

 

3^(-6x+4)*3^-x=3^(-6x-3)  

At this point, we can add the exponents on the left side of the equation because they now have common bases.

3^[(-6x+4)+(-x)]=3^(-6x-3)   Apply the Product Rule by adding the exponents when bases are equal.
3^(-7x+4)=3^(-6x-3)  

Clearly, we can set the powers of both sides of the equation equal to each other.

-7x+4=-6x-3   This results to a simple multistep equation.
x=7   So we add 6x first on both sides. Then, subtract by 4. And finally, divide by -1 to fully isolate x by itself!

The answer is x = 7. Easy!

 


Example 6: Solve the exponential equation 32^(x^2+1)*8^(x-5)=16^2x.

Solution:

32^(x^2+1)*8^(x-5)=16^2x   Given
(2^5)^(x^2+1)*(2^3)^(x-5)=(2^4)^2x  

Express each number with a base of 2.

Next, multiply the inner exponents to outer exponents using the Power to a Power Rule.

2^(5x^2+5)*2^(3x-15)=2^8x  

To generate a single base on the left side, use the Product Rule - copy the common base 2 and add the exponents.

2^[(5x^2+5)+(3x-15)]=2^8x   This is when we apply the Product Rule.
2^(5x^2+3x-10)=2^8x  

After the addition of exponents, we have single bases on each side.

It's time to set the powers equal to each other.

5x^2+3x-10=8x  

After equating the powers, we arrive at this quadratic equation.

We need to move all terms on one side while forcing the opposite side equal to zero.

5(x-2)(x+1)=0   Solve the quadratic equation using factoring method.
x=2 or x=-1   Using the Zero Property, we get these values for x.

The correct answers are x = 2 and x = −1.


Example 7: Solve the exponential equation (1/49)^-6x*49=√(7)          .

Solution:

(1/49)^-6x*49=√(7)            Given
(1/7^2)^-6x*7^2=√(7)           

Express each number as exponential number with a base of 7.

 

[7^-2]^-6x*7^2=(7)^(1/2)  

Apply the Negative Exponent Property on the left side.

Also, the square root symbol can be rewritten as exponent of 1/2.

7^12x*7^2=(7)^(1/2)   Apply the Power to a Power Rule on the left side.
7^(12x+2)=(7)^(1/2)   Express the left side with a single base using the Product Rule by copying the common base and adding the exponents.
12x+2=1/2   We can now set the powers equal to each other, then solve.
12x+2-2=(1/2)-2   To solve for x, subtract both sides by 2.
12x=-3/2   Simplify
x=-1/8   To finish this off, divide both sides by 12.

The final answer is x = -1/8.


 

Example 8: Solve the exponential equation [(125)^(-4x+5)]/[(25)^(x^2-5x+1)]=5.

Solution:

[(125)^(-4x+5)]/[(25)^(x^2-5x+1)]=5   Given
[5^3]^(-4x+5)/[5^2]^(x^2-5x+1)=5  

Express the numbers using the base 5.

Next, multiply the inner and outer exponents using the Power to a Power Rule.

 

[5]^(-12x+15)/[5]^(2x^2-10x+2)=5  

It looks like that we can use the Quotient Rule because we have the same bases on the numerator and denominator.

 

5^(-12x+15)-(2x^2-10x+2)=5   Subtract the exponent on the numerator by the one in the denominator.
5^-2x^2-2x+13=5^1  

Simplify

It's okay now to set the powers equal to each other, and then solve the equation.

-2x^2-2x+13=1   Solve the quadratic equation using factoring method.
x=-3 or x=2   Using the Zero Product Property, we obtain these values of x.

 

The final answers are x = −3 and x = 2.

 


 

Practice Problems with Answers
Worksheet 1 Worksheet 2

 

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