Solving Equations with Exponents and Logarithms: A Comprehensive Guide

Mathematics, especially algebra and calculus, involves a wide range of techniques for solving equations. One common type of equation involves exponents and logarithms. This guide provides a step-by-step approach to solving the equation (sqrt{2sqrt{3}}^x sqrt{2sqrt{3}}^x 2^x), using both algebraic manipulation and logarithmic properties.

Step-by-Step Solution

To solve the equation (sqrt{2sqrt{3}}^x sqrt{2sqrt{3}}^x 2^x), we first need to simplify the left-hand side. Let's start by observing the structure of the expression:

(sqrt{2sqrt{3}}^x sqrt{2sqrt{3}}^x 2^x)

Notice that we have two identical terms on the left-hand side, so we can rewrite it as:

(2sqrt{2sqrt{3}}^x 2^x)

Next, we can divide both sides by 2, assuming (x eq 0) (since (2^x eq 0)):

(sqrt{2sqrt{3}}^x 2^{x-1})

Now, we can further simplify (sqrt{2sqrt{3}}). We know that:

(sqrt{2sqrt{3}} sqrt{2 cdot sqrt{3}} sqrt{2 cdot sqrt{3}})

Let's rewrite the equation:

(sqrt{2sqrt{3}}^x 2^{x-1})

To proceed, we can take the logarithm of both sides to make it easier to solve for (x):

(x cdot log(sqrt{2sqrt{3}}) (x-1) cdot log(2))

This expands to:

(x cdot frac{1}{2} log(2sqrt{3}) x cdot log(2) - log(2))

Let's rearrange to isolate (x):

(x cdot frac{1}{2} log(2sqrt{3}) - x cdot log(2) -log(2))

Factor out (x):

(x left(frac{1}{2} log(2sqrt{3}) - log(2)right) -log(2))

Now, we can isolate (x):

(x frac{-log(2)}{frac{1}{2} log(2sqrt{3}) - log(2)})

To simplify further, let's compute (frac{1}{2} log(2sqrt{3}) - log(2)):

(frac{1}{2} log(2sqrt{3}) - log(2) frac{1}{2} log(2sqrt{3}) - frac{2}{2} log(2) frac{1}{2} log(2sqrt{3}) - frac{1}{2} log(4) frac{1}{2} left(log(2sqrt{3}) - log(4)right))

Thus, the expression becomes:

(x frac{-2 log(2)}{log(2sqrt{3}) - log(4)})

Using the properties of logarithms:

(log(2sqrt{3}) - log(4) logleft(frac{2sqrt{3}}{4}right))

So the final expression for (x) becomes:

(x frac{-2 log(2)}{logleft(frac{2sqrt{3}}{4}right)})

This gives us the solution in terms of logarithms. For a numerical approximation, you can calculate the values of the logarithms and compute (x) accordingly.

Alternative Approach

Another elegant approach involves recognizing the structure of the equation:

(2times sqrt{2sqrt{3}}^x 2times sqrt{frac{42sqrt{3}}{2}}^x 2times frac{sqrt{31}}{sqrt{2}}^x sqrt{31}^x cdot 2^{1-x/2} 2^{1-frac{x}{2} log_2{31}})

This simplifies to:

(1 - frac{x}{2} log_2{31} x)

Solving for (x), we get:

(x frac{1}{frac{3}{2} - log_2{31}})

Conclusion

We have demonstrated two different methods to solve the equation (sqrt{2sqrt{3}}^x sqrt{2sqrt{3}}^x 2^x). Both methods involve algebraic manipulation and logarithmic properties. Understanding these techniques can help you tackle more complex equations involving exponents and logarithms.

Keywords: equation solving, logarithms, exponentiation