Solve Algebra 2 Problem 43: 7^3x/2^7x-1=(43.2)^x+2

In summary, to solve an algebraic equation, you should isolate the variable on one side of the equal sign by using the laws of exponents and logarithms. It is also recommended to simplify the equation before using a calculator and to double-check your answer by plugging it back into the original equation. Common mistakes to avoid include forgetting the order of operations and incorrectly entering the problem into a calculator.
  • #1
mustang
169
0
Problem 43.
Solve and check.
7^3x divided by 2^7x-1 = (43.2)^x+2
 
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  • #2
I am assuming this is that you meant
[tex]\frac{7^{3x}}{2^{7x-1}}= 43.2^{x+2}[/tex]
rather than
[tex]\frac{7^3x}{2^7x-1}= 43.2^x+ 2[/tex]
which would be harder.

The key step is to use the logarithm to get rid of the "exponentials" (It really doesn't matter which base logarithm you use):
[tex]log(\frac{7^{3x}}{2^{7x-1}})= log(43.2^{x+2})[/tex]
which is:
[tex] log(7^{3x})- log(2^{7x-1})= log(34.2^{x+2})[/tex]
[tex] 3x log(7)- (7x-1)log(2)= (x+2)log(34.2)

That's now a simple linear equation with some peculiar coefficients.
 
  • #3


To solve this problem, we will first use the properties of exponents to rewrite the equation in a simpler form. We know that (a^m)^n = a^(m*n) and (a/b)^n = a^n/b^n. Using these properties, we can rewrite the equation as:

(7^3)^x / (2^7)^x-1 = (43.2)^x+2

Now, we can simplify further by evaluating the exponents on the left side of the equation:

7^(3x) / 2^(7x-1) = (43.2)^x+2

Next, we can use the power rule of exponents to rewrite 43.2 as (432/10) and raise both sides of the equation to the power of x+2:

7^(3x) / 2^(7x-1) = (432/10)^(x+2)

Using the property of exponents (a^m)^n = a^(m*n), we can rewrite the right side of the equation as:

7^(3x) / 2^(7x-1) = (432/10)^x * (432/10)^2

Now, we can simplify the right side of the equation by evaluating (432/10)^2 as 432^2/10^2:

7^(3x) / 2^(7x-1) = (432/10)^x * (432^2/10^2)

Next, we can use the power rule of exponents again to rewrite the right side of the equation as:

7^(3x) / 2^(7x-1) = (432/10)^x * 432^(2x)/10^(2x)

Finally, we can use the property of exponents (a/b)^n = a^n/b^n to rewrite the right side of the equation as:

7^(3x) / 2^(7x-1) = (432/10 * 432^(2x)) / (10 * 10^(2x))

Now, we can simplify the right side of the equation by combining like terms:

7^(3x) / 2^(7x-1) = (432 * 432^(2x)) / (10 * 10^(2x+1))

To solve for x, we can take the natural logarithm of both sides of
 

1. How do I solve this algebraic equation?

To solve this equation, you need to isolate the variable on one side of the equal sign. In this case, you can start by dividing both sides by 7^3x to get rid of it on the left side. Then, you can simplify the right side by using the laws of exponents to get rid of the parentheses. Finally, you can use logarithms to solve for x.

2. Can I use a calculator to solve this problem?

Yes, you can use a calculator to help you solve this problem. However, you will still need to know how to use the laws of exponents and logarithms to simplify the problem before using the calculator to get the final answer.

3. What is the best way to approach solving this problem?

The best way to approach solving this problem is to first simplify the equation by using the laws of exponents. Then, use logarithms to isolate the variable and solve for x. Make sure to check your answer by plugging it back into the original equation.

4. Are there any tips for solving algebraic equations like this?

One tip for solving algebraic equations is to always start by simplifying the equation as much as possible before solving for the variable. This can make the problem more manageable and easier to solve.

5. What are some common mistakes to avoid when solving this type of problem?

One common mistake to avoid when solving this type of problem is forgetting to use the correct order of operations. Remember to always simplify the equation before using logarithms to isolate the variable. Also, be careful when using a calculator to make sure you enter the problem correctly and get the correct final answer.

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