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.
Key Steps in Solving Exponential Equations without Logarithms
Make the base on 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 on 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 Property
2) Negative Exponent Property
3) Product Rule
4) Quotient Rule
5) Power to a Power Rule
Let’s take a look at some examples!
Examples of How to Solve Exponential Equations without Logarithms
Example 1: Solve the exponential equation below using the Basic Properties of Exponents.
- Express the denominator of the right side with a base of 5. We have 125 = 53.
Apply the Negative Exponent Property.
- At this point, the bases are the same therefore set the powers equal to each other.
- This is just a simple one-step linear equation.
- To solve for x, divide both sides by 3. That’s it!