Solve for: 4^x+4^(1/x)=18

just by inspection, x=2

As well:

x = 1/2

To solve the equation 4^x + 4^(1/x) = 18, we can follow these steps:

Step 1: Let's rewrite the equation using a common base. Since both terms have base 4, we can rewrite 18 as 4^(log4(18)), using the logarithm base-change rule.

Therefore, our equation becomes:

4^x + 4^(1/x) = 4^(log4(18))

Step 2: Now, we can equate the exponents of the same base. This means:

x = log4(18)

Step 3: To solve for x, we need to evaluate log4(18). We can rewrite the logarithmic equation as an exponential equation:

4^x = 18

Step 4: We can now solve for x by taking the logarithm of both sides of the equation using the base 4 logarithm:

log4(4^x) = log4(18)

x = log4(18)

Step 5: We can use a calculator to evaluate log4(18). The base 4 logarithm is equivalent to the common logarithm divided by log4. Therefore:

x ≈ log(18) / log(4)

Using a calculator, we find:

x ≈ 1.8019

So, the approximate solution to the equation is x ≈ 1.8019.