Mastering Exponential Equations: Techniques On How To Solve Exponential Equations With Different Bases And Exponents

Exponential equations with different bases and exponents can be intimidating to solve at first glance, but with the right understanding and approach, they can be tackled with ease. In this article, we will go through the steps and techniques needed to solve exponential equations with different bases and exponents.

Solving Exponential Equations with Same Base:

Before we dive into solving exponential equations with different bases, it’s important to understand how to solve equations with the same base. The key idea is to use the property of logarithms that states, “log_a (x^m) = m log_a(x)”.

For example, let’s say we have the equation 2^x = 16. We can rewrite this as log_2(16) = x, since log_2 (16) = 4. Thus, x = 4.

Solving Exponential Equations with Different Bases:

When we have different bases, we can use the change of base formula to rewrite the equation in terms of a common base. The change of base formula is as follows:

log_a(x) = (log_b(x))/(log_b(a))

For example, if we have the equation 3^x = 5^2, we can rewrite this as x log_3(3) = 2 log_3(5) using the change of base formula. Since log_3(3) = 1, we can simplify this to x = 2 log_3(5). Using a calculator, we can approximate this to x = 2.37.

Solving Exponential Equations with Different Exponents:

When we have different exponents, we need to manipulate the equation so that the exponents are the same. One technique is to use the fact that a^b / a^c = a^(b-c).

For example, let’s say we have the equation 2^x = 8^(x-1). We can rewrite this as 2^x = (2^3)^(x-1), since 8 is equal to 2^3. This gives us 2^x = 2^(3x-3). From here, we can equate the exponents and solve for x.

x = 3x – 3 2x = 3 x = 1.5

Solving Exponential Equations with Both Different Bases and Exponents:

When we have both different bases and exponents, we need to use a combination of the techniques mentioned above. Let’s look at an example.

2^x = 3^(2x-1)

Using the change of base formula, we can rewrite this as x log_2(2) = (2x-1) log_2(3). Simplifying this, we get x = (2 log_2(3))/(2-log_2(2)). Using a calculator, we can approximate this to x = 2.41.