Systems of Exponential and Logarithmic Equations

Logarithms are another way of writing exponents in mathematics. A number’s logarithm with a base equals another number. Exponentiation is the inverse function of a logarithm. If 10² = 100, for example, log10 100 = 2.

As a result, we may deduce that

logbx = n or bn = x

The base of the logarithmic function is b.

“The logarithm of x to the base b is equal to n,” it says.

We will cover the definition of logarithms, the two types of logarithms (common and natural logarithms), and other properties of logarithms with numerous solved cases in this post.

Definition

The power to which a number must be increased to obtain additional values is defined as a logarithm. It is the most practical method of expressing enormous numbers. Multiplication and division of logarithms can also be stated in the form of logarithms of addition and subtraction, thanks to a number of important features of logarithms.

“The exponent by which b must be raised to give an is the logarithm of a positive real number a with respect to base b, a positive real number not equal to 1.”

i.e. by= a ⇔logba=y

Where,

The numbers “a” and “b” are both positive real numbers.

Y is an integer.

“a” stands for argument and is located inside the log. “b” stands for base and is located at the bottom of the log.

In other terms, the logarithm answers the question, “How many times does a number have to be multiplied to get the other number?”

Exponential Function

An exponential function is a mathematical function of the form f (x) = ax, where “x” is a variable and “a” is a constant that is the function’s base and must be greater than 0. The transcendental number e, which is approximately equal to 2.71828, is the most often used exponential function basis.

Systems of Logarithmic Equations by Elimination

Multiple equations and unknown variables make up a system of equations. Because each equation cannot be solved independently, systems of equations can be difficult to work with, but there are certain strategies that can assist!

You must combine the equations to obtain an equation with only ONE unknown variable to solve any system of equations, not simply one including logarithmic equations. There are a few popular approaches to this.

In some circumstances, you can start by using the elimination approach. You combine the two equations in this manner to eliminate one of the variables. Let’s explore how that works using a logarithmic equation system.

In the logarithmic equations below, how would you solve for x and y?

log(x)+ log(y) =1

log(10x) – log(y) = 2

First, observe that by combining these two equations, the variable y is eliminated. You’ll be able to solve for x after that.

log(x)+ log(y) =1

After you’ve eliminated y, you can solve for x using logarithmic addition rules:

log(x) + log (10x) = 3

log(10x²) = 3

10x² = 10³

x² = 100

x = ±10

As you can see, there are two options available. X can be either ten or ten. What criteria do you use to determine which is correct? Some solutions to logarithmic equations may not be attainable, thus you should always double-check them. To test a solution, try applying it to the original problem and seeing if it works. Let’s have a look at what happens when x = 10 and x = -10.

log(10) + log (10•10) = 3

log(–10) + log (10•–10) = 3

As a result, the only valid answer to this equation is 10. Finally, in any of the two equations in the original system, you can determine y by replacing x with 10.

log(10) + log(y) =3

1 + log(y) = 3

log(y) = 2

y = 10² = 100.

System of Exponential Equations

There are a few different approaches to solving a system of exponential equations. The following are the strategies:

• Using a mix of techniques such as system of linear equations and exponent laws

• Solving the system using elimination, substitution, or comparison as you would a regular system of linear equations

The technique should be used in accordance with the equations’ requirements. Keep in mind that you can only solve a system of exponential equations if the bases of two or more of them are the same. If the bases are the same, the exponential system of equations can be solved by equalizing the exponents on the left and right sides of the equations.

However, you may be asked to solve a system of exponential equations with different bases in the future. In this case, you should examine if you can use the exponent rules to set the identical bases for both equations. 

Exponential Solved Examples

Example1.  the exponential function simpler. 2ˣ – 2ˣ⁺¹.

Solution. 2ˣ – 2ˣ⁺¹ is an exponential function.

Using the formula aˣ ay = ax+y

2x+1 can so be written as 2ˣ. 2

As a result, the following function is written:

2ˣ-2ˣ⁺¹ = 2ˣ-2ˣ. 2

Remove the word 2x now.

2ˣ-2ˣ⁺¹ = 2ˣ-2ˣ. 2 = 2ˣ(1-2)

2ˣ-2ˣ⁺¹ = 2ˣ(-1)

2ˣ-2ˣ⁺¹ = – 2ˣ

As a result, the above exponential function 2ˣ – 2ˣ⁺¹ is simplified to – 2ˣ.

Example2. (¼)ˣ = 64 is the exponential equation to solve.

Solution. The following is an exponential equation:

(¼)ˣ = 64

We get: (a/b)ˣ = aˣ/bˣ using the exponential method.

1ˣ/4ˣ = 43

1/4ˣ  = 4³ [since 1ˣ = 1]

(1)(4⁻ˣ) = 4³

4⁻ˣ = 4³

The bases are all the same.

As a result of equating our powers;

x = -3.

Conclusion

The ability of logarithms to solve exponential equations accounts for much of their potency. Sound (decibel measurements), earthquakes (Richter scale), star brightness, and chemistry are all examples of this (pH balance, a measure of acidity and alkalinity). 

Earthquake magnitude is calculated using logarithms by seismologists. Logarithms are used by financial institutions to calculate the length of loan repayments. The rate of radioactive decay is calculated using logarithms. Logarithms are used by biologists to compute population growth rates.