# Solve 8^x+13^x=108: Step-by-Step Algebraic Guide

• beethoven'smahomeboy
In summary, the conversation is about solving the equation 8^x+13^x=108 for x. The participants discuss different methods to solve it, but it is determined that there is no exact solution and the only way to solve it is by approximation using methods like Newton-raphson.
beethoven'smahomeboy
Help me solve for x in detail please:

8^x+13^x=108

Simply done graphically, but algebraically?

It can't be solved with logs or anything like that.

I figured as much after working with it for a while, but I thought I might have just been missing something. How does one solve it then?

You can only approximate the solution, which is:

$$x \approx 1.682790571570192534987737142973064792462$$

I think you can't do it,without a calculator.Maple returns $\left\{ x=1.\,68279\,0572\right\}$,but it can't be solved algebraically.In fact,most transcendental equations can't be solved.

Daniel.

you can solve it by using Newton-raphson method

quentinchin said:
you can solve it by using Newton-raphson method

Only approximate, not an exact solution. There is no exact solution method.

## 1. How do I solve 8^x+13^x=108?

To solve this equation, we will use logarithms. First, we rewrite the equation as 8^x=108-13^x. Then, take the logarithm of both sides using any base. We will use the natural logarithm (ln) for this example. So, the equation becomes ln(8^x)=ln(108-13^x). Using the power rule of logarithms, we can rewrite the left side as xln(8). We then use the logarithm property that states ln(a-b)=ln(a)-ln(b) to rewrite the right side as ln(108)-ln(13^x). Now, we have xln(8)=ln(108)-xln(13). Finally, we solve for x by dividing both sides by ln(8) and then factoring out x. The final solution is x=ln(108)/ln(8)-ln(13).

## 2. Can I use a different base for the logarithm?

Yes, you can use any base for the logarithm. However, using a base that is a power of one of the bases in the equation will make the calculation simpler. In this case, we used ln because it is a power of e, which is the base of the natural logarithm.

## 3. How do I solve for x if the equation has a different constant on the right side?

If the equation is in the form a^x+b^x=c, where a, b, and c are constants, the steps are similar. You will still use logarithms to rewrite the equation and then solve for x by dividing both sides by the logarithm of the base used. The solution will be in terms of the constants and the base used for the logarithm.

## 4. Can I use a calculator to solve this equation?

Yes, you can use a calculator to solve this equation. However, make sure to use the appropriate base for the logarithm function on your calculator. Also, be careful with rounding errors, as they can affect the final solution.

## 5. Are there any other methods to solve this equation?

Yes, there are other methods to solve this equation, such as using graphing or numerical methods. However, using logarithms is the most efficient and straightforward method for solving this type of equation.

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