Step-by-Step Solution for Solving Exponential Equations with Fractional Bases

When you have two equations that are "the same" except for different values, you can often relate them by viewing one as a special case of the other. In this problem you are not really doing that. You are just saying that you know a certain special case, and you are hoping that someone will build on that special case and help you generalize it to solve the problem in question. So, there is no need to have 4p = 3p-1, etc. in the equations here. That was just what you knew how to do for some other problem.In this problem, you can convert some terms to powers of 9:(1/9)^m = 81^m +
  • #1
Corkery
32
0

Homework Statement


Solve the equation:
(1/9)^m = 81^m + 4


Homework Equations


I know how to do a similar equation:
9^2p = 27^p-1

3^4p = 3^3p-1

cancel out the 3's because the base is the same.

4p = 3p-1
-3p -3p

1p = -1


The Attempt at a Solution


I know how to do ones like that but when it comes to this I start to struggle with fractions. :'(.

I'm totally lost and can't even make an attempt to this equation. I'm not asking for the answer but just a maybe the first step or something to get me on the right path.
 
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  • #2
How about converting some terms to powers of 9?
 
  • #3
Corkery said:

Homework Statement


Solve the equation:
(1/9)^m = 81^m + 4


Homework Equations


I know how to do a similar equation:
9^2p = 27^p-1

3^4p = 3^3p-1

cancel out the 3's because the base is the same.

4p = 3p-1
-3p -3p

1p = -1


The Attempt at a Solution


I know how to do ones like that but when it comes to this I start to struggle with fractions. :'(.

I'm totally lost and can't even make an attempt to this equation. I'm not asking for the answer but just a maybe the first step or something to get me on the right path.

[tex] (\frac{1}{9})^m = 81^{m+4} [/tex]

[tex] (9^{-1})^m = (9^2)^{m+4} [/tex]

[tex] (9)^{-m} = (9)^{2m+8} [/tex]

[tex] -m = 2m+8 [/tex]

[tex] -8 = 3m [/tex]

[tex] m = - \frac{8}{3} [/tex]
 
  • #4
Corkey,

The way you manage your notation is very important: you wrote
(1/9)^m = 81^m + 4


2. Homework Equations
I know how to do a similar equation:
9^2p = 27^p-1

3^4p = 3^3p-1

cancel out the 3's because the base is the same.

4p = 3p-1
-3p -3p

1p = -1
which showed your original equation of difficulty to be very different from what you meant. In simple text-based expressiveness, you needed proper grouping saying (1/9)^m = 81^(m+4); The way you first wrote it, the "4" was not included as part of the exponent.
 

1. How do you solve an exponential equation with a fractional base?

To solve an exponential equation with a fractional base, you need to use logarithms. Take the logarithm of both sides of the equation and then use logarithm properties to simplify the equation into a form where you can easily solve for the variable.

2. What is the general formula for solving exponential equations with fractional bases?

The general formula for solving exponential equations with fractional bases is:
logb(x) = y, where b is the base, x is the exponent, and y is the solution. To solve for x, you would take the logarithm of both sides using base b.

3. Can you explain the steps involved in solving an exponential equation with a fractional base?

First, take the logarithm of both sides of the equation using a base that will simplify the equation. Then, use logarithm properties to simplify the equation into a form where you can easily solve for the variable. Finally, solve for the variable and check your solution by substituting it back into the original equation.

4. What are the common mistakes to avoid when solving exponential equations with fractional bases?

One common mistake is forgetting to take the logarithm of both sides of the equation. Another mistake is using the wrong base when taking the logarithm, which can lead to an incorrect solution. It is also important to check your solution by substituting it back into the original equation.

5. Are there any tips or tricks for solving exponential equations with fractional bases?

One helpful tip is to choose a base that will simplify the equation, such as using a base of 2 for fractional bases of 1/2 or 1/4. It is also important to remember logarithm properties, such as logb(xa) = a*logb(x). And as always, double-check your solution to ensure it is correct.

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