How Can You Solve the Equation 3^x + x = 4?

[nineaxe]

Homework Statement

3^x + x = 4

solve for x.

Homework Equations

I'm thinking of using the logarithm laws. log(a.b) = log a + logb log(a/b) = log a - log b

log(a^b) = bloga

The Attempt at a Solution

Well if I isolate the exponential
3^x = 4-x

take logs on both sides

xlog3 = log(4-x)

I can't seem to get the x out of the log.

 

[Cyosis]

You won't be able to find a solution algebraically. However the value for x is quite obvious. Take a good look at the equation in its original form.

 

[Architeuthis Dux]

To solve this equation, you can use the change of base formula for logarithms. This states that loga(b) = logc(b)/logc(a), where c is any base. In this case, we can choose c = 3 since it is the base of the exponential in the equation. So, we have:

xlog3 = log(4-x)

x = log(4-x)/log3

Now, we can use the properties of logarithms to simplify the logarithmic expression on the right side. Remember that loga(b) = c is equivalent to a^c = b. Using this, we can rewrite the equation as:

x = (4-x)^log3

Finally, we can use algebraic methods to solve for x. One approach is to expand the exponent on the right side using the power rule, giving us:

x = 4^log3 - x^log3

Then, we can rearrange the terms to get all the x terms on one side:

x + x^log3 = 4^log3

Factor out the x on the left side:

x(1 + x^(log3 - 1)) = 4^log3

Divide both sides by 1 + x^(log3 - 1):

x = 4^log3 / (1 + x^(log3 - 1))

And there you have the solution for x. You can plug in the numbers for log3 and simplify further if needed.

 

[Architeuthis Dux]

To solve this equation, we can use the logarithm laws as you suggested. However, we need to use the change of base formula to get rid of the logarithm with base 3. The change of base formula states that logb(x) = loga(x)/loga(b).

Therefore, we can rewrite the equation as:

x = log(4-x)/log3

This is still not a simple equation to solve, but we can use numerical methods such as graphing or using a calculator to approximate the value of x. We can also use iterative methods such as the Newton-Raphson method to find a more accurate solution.

Another approach would be to use algebraic manipulation to rewrite the equation in a simpler form. For example, we can rewrite 3^x as (3^2)^x, and then use the property (a^b)^c = a^(b*c) to simplify the equation to:

(3^2)^x + x = 4

9^x + x = 4

We can then use trial and error to find the value of x that satisfies this equation. For example, we can try x = 1, which gives us 9 + 1 = 10. This is greater than 4, so we know that the solution is less than 1. We can then try x = 0.5, which gives us 3 + 0.5 = 3.5. This is still greater than 4, so we know that the solution is between 0.5 and 1. We can continue this process and narrow down the range until we find a value of x that satisfies the equation.

In conclusion, solving exponential equations can be challenging and may require the use of logarithm laws and numerical methods. It is important to carefully manipulate the equation and use trial and error to find the solution.