Solving the Equation (3^x 4^x 5^x - 6^x 0) Algebraically

Algebraically solving the equation (3^x 4^x 5^x - 6^x 0) involves several steps, from simplifying the equation to defining a function and numerically approximating the solution. In this article, we will discuss this process in detail.

1. Simplifying the Equation

The given equation is:

$$3^x 4^x 5^x - 6^x 0$$

First, we can rearrange it to:

$$3^x 4^x 5^x 6^x$$

Next, divide all terms by (6^x):

$$frac{3^x}{6^x} frac{4^x}{6^x} frac{5^x}{6^x} 1$$

Using the properties of exponents, we can rewrite this as:

$$left(frac{3}{6}right)^x left(frac{4}{6}right)^x left(frac{5}{6}right)^x 1$$

Simplifying the fractions, we get:

$$left(frac{1}{2}right)^x left(frac{2}{3}right)^x left(frac{5}{6}right)^x 1$$

2. Defining and Analyzing the Function

Define the function:

$$f(x) left(frac{1}{2}right)^x left(frac{2}{3}right)^x left(frac{5}{6}right)^x$$

We seek to find (x) such that (f(x) 1).

Behavior of (f(x))

As (x rightarrow -infty): Each term (left(frac{1}{2}right)^x), (left(frac{2}{3}right)^x), and (left(frac{5}{6}right)^x) approaches infinity, so (f(x) rightarrow infty). As (x rightarrow infty): Each term approaches 0, so (f(x) rightarrow 0).

Since (f(x)) is continuous and decreases from (infty) to 0, there is at least one root where (f(x) 1).

3. Numerical Approximation

While an exact algebraic solution is complex, numerical methods or graphing techniques can help find approximate solutions. We will test some values to find where (f(x) 1).

Testing Values

(x 0): (left(frac{1}{2}right)^0 left(frac{2}{3}right)^0 left(frac{5}{6}right)^0 1 1 1 3 > 1) (x 1): (left(frac{1}{2}right)^1 left(frac{2}{3}right)^1 left(frac{5}{6}right)^1 frac{1}{2} frac{2}{3} frac{5}{6} frac{9}{6} frac{4}{6} frac{5}{6} frac{18}{6} 3 > 1) (x 2): (left(frac{1}{2}right)^2 left(frac{2}{3}right)^2 left(frac{5}{6}right)^2 frac{1}{4} frac{4}{9} frac{25}{36})

Using a common denominator of 36:

(frac{9}{36} frac{16}{36} frac{25}{36} frac{50}{36} approx 1.39 > 1) (x 3): (left(frac{1}{2}right)^3 left(frac{2}{3}right)^3 left(frac{5}{6}right)^3 frac{1}{8} frac{8}{27} frac{125}{216})

Using a common denominator of 216:

(frac{27}{216} frac{64}{216} frac{125}{216} frac{216}{216} 1)

Thus, we find that (x 3) is a solution to the equation (3^x 4^x 5^x - 6^x 0).