1. ## Finding exponential variables with subtracting or addition

Okay story is my teacher assigned us work in class after we learned how to solve exponential equations (with variables as exponents and multiplication signs between bases). This was way too easy for me and so Idid it but I saw other questions with subtration and addition so I was curious and tried it. But I can't figure out how to even start it off. This is for my own entertainment not for marks.

Example of a question,

4^x + 4^(x+1) = 40

So what I tried was expanding 4^(x+ 1) into 4(4^x) and I for some reason still couldn't get it. I used my guess-ta-mation skills and got x= 1.5 but I lack the steps to get there. I would also like to avoid log.

Thanks for trying if you do.

2. ## Re: Finding exponential variables with subtracting or addition

Originally Posted by Ravagence
Okay story is my teacher assigned us work in class after we learned how to solve exponential equations (with variables as exponents and multiplication signs between bases). This was way too easy for me and so Idid it but I saw other questions with subtration and addition so I was curious and tried it. But I can't figure out how to even start it off. This is for my own entertainment not for marks.

Example of a question,

4^x + 4^(x+1) = 40

So what I tried was expanding 4^(x+ 1) into 4(4^x) and I for some reason still couldn't get it. I used my guess-ta-mation skills and got x= 1.5 but I lack the steps to get there. I would also like to avoid log.

Thanks for trying if you do.
Expanding $4^{x+1} = 4 \cdot 4^x$ is a good start. Now you can collect like terms: $1 \cdot 4^x + 4 \cdot 4^x = 5 \cdot 4^x$

Overall you have $5 \cdot 4^x = 40 \Leftrightarrow 4^x = 8$

Since you don't want to use logs we can use the fact that both 4 and 8 are powers of 2 and so: $2^{2x} = 2^3$ and since the bases are equal so too must be the exponents: $2x = 3$

3. ## Re: Finding exponential variables with subtracting or addition

Originally Posted by e^(i*pi)
Expanding $4^{x+1} = 4 \cdot 4^x$ is a good start. Now you can collect like terms: $1 \cdot 4^x + 4 \cdot 4^x = 5 \cdot 4^x$

Overall you have $5 \cdot 4^x = 40 \Leftrightarrow 4^x = 8$

Since you don't want to use logs we can use the fact that both 4 and 8 are powers of 2 and so: $2^{2x} = 2^3$ and since the bases are equal so too must be the exponents: $2x = 3$
Thank you so much! So I was on the right track all along. I just didn't use the 1(4^x) so I ended having 4^x = 10 instead.

Another quick question, is there anything to keep in mind when using this method on other equations? Or is it pretty straightforward?

4. ## Re: Finding exponential variables with subtracting or addition

Originally Posted by Ravagence
Thank you so much! So I was on the right track all along. I just didn't use the 1(4^x) so I ended having 4^x = 10 instead.

Another quick question, is there anything to keep in mind when using this method on other equations? Or is it pretty straightforward?

Try to factor or combine like terms wherever possible. I usually find that "splitting" an exponent like you did is a positive step in solving an equation like this but if it ever isn't it's easy enough to put back together.

5. ## Re: Finding exponential variables with subtracting or addition

Originally Posted by e^(i*pi)
Try to factor or combine like terms wherever possible. I usually find that "splitting" an exponent like you did is a positive step in solving an equation like this but if it ever isn't it's easy enough to put back together.
Thank you very much.