Solving Exponential Equations A Step By Step Guide

In this comprehensive guide, we will explore the methods to solve various exponential equations. Exponential equations are mathematical expressions where the variable appears in the exponent. Solving these equations often requires a solid understanding of exponent rules and algebraic manipulation. We'll dissect several examples, providing step-by-step solutions and clear explanations to enhance your understanding. This article will cover equations such as 3 \cdot 2^{x+2} - 5 \cdot 2^{x+1} - 2^x = 2, 5^{x+2} - 2 \cdot 5^{x+1} - 12 \cdot 5^x = 375, 2^x \cdot 3^{x+1} = 108, 3^{x+1} \cdot 5^x = 675, and 3^{x+2} \cdot 7^x - 3^x \cdot 7^{x+1} = 2940. By the end of this guide, you will be well-equipped to tackle a wide range of exponential equations.

g) Solve: 3 \cdot 2^{x+2} - 5 \cdot 2^{x+1} - 2^x = 2

To solve the exponential equation 3 \cdot 2^{x+2} - 5 \cdot 2^{x+1} - 2^x = 2, the key is to simplify the equation by expressing each term with a common base and exponent. We can rewrite the terms using exponent rules. Specifically, we recall that $a^{m+n} = a^m \\cdot a^n$. Applying this rule, we can break down the terms as follows:

• $2^{x+2} = 2^x \\cdot 2^2 = 4 \\cdot 2^x$
• $2^{x+1} = 2^x \\cdot 2^1 = 2 \\cdot 2^x$

Substituting these back into the original equation gives us:

$3 \\cdot (4 \\cdot 2^x) - 5 \\cdot (2 \\cdot 2^x) - 2^x = 2$

This simplifies to:

$12 \\cdot 2^x - 10 \\cdot 2^x - 2^x = 2$

Now, we can factor out $2^x$:

$2^x (12 - 10 - 1) = 2$

Which further simplifies to:

$2^x (1) = 2$

So, we have:

$2^x = 2$

Since $2^1 = 2$, we conclude that:

$x = 1$

Thus, the solution to the exponential equation 3 \cdot 2^{x+2} - 5 \cdot 2^{x+1} - 2^x = 2 is x = 1. This method of simplifying and factoring out the common exponential term is crucial for solving many exponential equations. Always remember to double-check your work by substituting the solution back into the original equation to ensure accuracy.

h) Solve: 5^{x+2} - 2 \cdot 5^{x+1} - 12 \cdot 5^x = 375

To tackle the exponential equation 5^{x+2} - 2 \cdot 5^{x+1} - 12 \cdot 5^x = 375, the initial step involves simplifying the terms using exponent rules. Recall the rule $a^{m+n} = a^m \\cdot a^n$. We can rewrite the equation as follows:

• $5^{x+2} = 5^x \\cdot 5^2 = 25 \\cdot 5^x$
• $5^{x+1} = 5^x \\cdot 5^1 = 5 \\cdot 5^x$

Substituting these back into the original equation, we get:

$25 \\cdot 5^x - 2 \\cdot (5 \\cdot 5^x) - 12 \\cdot 5^x = 375$

Simplifying further:

$25 \\cdot 5^x - 10 \\cdot 5^x - 12 \\cdot 5^x = 375$

Now, factor out $5^x$:

$5^x (25 - 10 - 12) = 375$

This simplifies to:

$5^x (3) = 375$

Divide both sides by 3:

$5^x = 125$

Since $125 = 5^3$, we have:

$5^x = 5^3$

Therefore, the solution is:

$x = 3$

Thus, the solution to the exponential equation 5^{x+2} - 2 \cdot 5^{x+1} - 12 \cdot 5^x = 375 is x = 3. This approach of breaking down exponential terms, factoring, and simplifying is fundamental in solving such equations. Always verify the solution by substituting it back into the original equation to ensure correctness.

i) Solve: 2^x \cdot 3^{x+1} = 108

To solve the exponential equation 2^x \cdot 3^{x+1} = 108, we first need to simplify the equation. Begin by using the exponent rule $a^{m+n} = a^m \\cdot a^n$ to rewrite $3^{x+1}$:

$3^{x+1} = 3^x \\cdot 3^1 = 3 \\cdot 3^x$

Substitute this back into the original equation:

$2^x \\cdot (3 \\cdot 3^x) = 108$

Rearrange the terms:

$3 \\cdot 2^x \\cdot 3^x = 108$

Divide both sides by 3:

$2^x \\cdot 3^x = 36$

Now, use the rule $a^x \\cdot b^x = (a \\cdot b)^x$:

$(2 \\cdot 3)^x = 36$

This simplifies to:

$6^x = 36$

Since $36 = 6^2$, we have:

$6^x = 6^2$

Therefore, the solution is:

$x = 2$

Thus, the solution to the exponential equation 2^x \cdot 3^{x+1} = 108 is x = 2. This method demonstrates the importance of recognizing and applying exponent rules to simplify equations and find solutions. Always ensure to check your solution by plugging it back into the original equation.

j) Solve: 3^{x+1} \cdot 5^x = 675

To solve the exponential equation 3^{x+1} \cdot 5^x = 675, the first step is to simplify the equation using exponent rules. We can rewrite $3^{x+1}$ using the rule $a^{m+n} = a^m \\cdot a^n$:

$3^{x+1} = 3^x \\cdot 3^1 = 3 \\cdot 3^x$

Substitute this back into the original equation:

$(3 \\cdot 3^x) \\cdot 5^x = 675$

Rearrange the terms:

$3 \\cdot 3^x \\cdot 5^x = 675$

Divide both sides by 3:

$3^x \\cdot 5^x = 225$

Now, apply the rule $a^x \\cdot b^x = (a \\cdot b)^x$:

$(3 \\cdot 5)^x = 225$

This simplifies to:

$15^x = 225$

Recognize that $225 = 15^2$, so:

$15^x = 15^2$

Therefore, the solution is:

$x = 2$

Thus, the solution to the exponential equation 3^{x+1} \cdot 5^x = 675 is x = 2. This process underscores the utility of simplifying terms and applying exponent rules to solve exponential equations effectively. Always remember to verify your solutions by substituting them back into the original equation.

k) Solve: 3^{x+2} \cdot 7^x - 3^x \cdot 7^{x+1} = 2940

To solve the exponential equation 3^{x+2} \cdot 7^x - 3^x \cdot 7^{x+1} = 2940, we begin by simplifying the terms using exponent rules. Rewrite $3^{x+2}$ and $7^{x+1}$ using the rule $a^{m+n} = a^m \\cdot a^n$:

• $3^{x+2} = 3^x \\cdot 3^2 = 9 \\cdot 3^x$
• $7^{x+1} = 7^x \\cdot 7^1 = 7 \\cdot 7^x$

Substitute these back into the original equation:

$(9 \\cdot 3^x) \\cdot 7^x - 3^x \\cdot (7 \\cdot 7^x) = 2940$

Rewrite the equation:

$9 \\cdot 3^x \\cdot 7^x - 7 \\cdot 3^x \\cdot 7^x = 2940$

Factor out $3^x \\cdot 7^x$:

$3^x \\cdot 7^x (9 - 7) = 2940$

Simplify:

$3^x \\cdot 7^x (2) = 2940$

Divide both sides by 2:

$3^x \\cdot 7^x = 1470$

Use the rule $a^x \\cdot b^x = (a \\cdot b)^x$:

$(3 \\cdot 7)^x = 1470$

This simplifies to:

$21^x = 1470$

Now, we can rewrite 1470 as $21^2 \\cdot \\frac{10}{3}$, but this doesn't lead to a straightforward solution. Instead, we should check our previous steps for any errors. Going back, we see the equation is correctly simplified to:

$21^x = 1470$

We made an error in simplifying 1470. Let's re-evaluate the step:

Divide both sides by 2:

$3^x \\cdot 7^x = 1470$

$(3 \\cdot 7)^x = 1470$

$21^x = 1470$

Now, we divide 2940 by 2 to get 1470. We need to express 1470 as a product involving powers of 21. Let's factor 1470:

$1470 = 2 \\cdot 3 \\cdot 5 \\cdot 7^2 = 21 \\cdot 70$

It seems there was an error in the problem statement or in our calculations, as 1470 is not a power of 21. However, if the equation was intended to be:

$3^{x+2} \\cdot 7^x - 3^x \\cdot 7^{x+1} = 2940$

and we correctly reached:

$2 \\cdot (3^x \\cdot 7^x) = 2940$

Then,

$3^x \\cdot 7^x = 1470$

$(21)^x = 1470$

This still doesn't lead to an integer solution. However, if the right side of the original equation was 294, then:

$2 \\cdot (3^x \\cdot 7^x) = 294$

$3^x \\cdot 7^x = 147$

$(21)^x = 147$

$147 = 3 \\cdot 7^2$, which is also not a power of 21.

Let's consider a possible intended equation where the right side was 4410. If:

$2 \\cdot 3^x \\cdot 7^x = 2 \\cdot (21)^x = 4410$

$(21)^x = 2205$, which is not a power of 21 either.

Upon further review, let's reconsider the original steps and factor 1470 correctly:

$1470 = 2 \\cdot 3 \\cdot 5 \\cdot 7^2 = 2 \\cdot 5 \\cdot (3 \\cdot 7^2) = 1029$

However, if we made a mistake earlier, and the correct equation is:

$2 \\cdot 3^x \\cdot 7^x = 2940$

Then:

$3^x \\cdot 7^x = 1470$

$(21)^x = 1470$

Let's assume there was a mistake and the original equation on the right side was a multiple of 21. If we had:

$21^x = 441 = 21^2$

Then $x = 2$. If we substitute $x=2$ back into the factored form:

$2 \\cdot (3^2 \\cdot 7^2) = 2 \\cdot (9 \\cdot 49) = 2 \\cdot 441 = 882$, which doesn't match 2940.

Therefore, without a clear integer solution, it indicates a potential issue with the original problem statement or a need for numerical methods to find an approximate solution. Due to the complexity and lack of an apparent straightforward solution, a direct numerical solution or verification of the original problem statement is advisable.

In this comprehensive guide, we have walked through the solutions to several exponential equations. The key to solving these equations lies in the skillful application of exponent rules, simplification techniques, and factoring. While some equations yield straightforward integer solutions, others may require more advanced methods or may even indicate a need to review the problem statement for errors. By understanding the core principles and practicing regularly, you can significantly enhance your ability to solve exponential equations effectively. Always remember to double-check your solutions by substituting them back into the original equations to ensure accuracy and completeness.