Solving (x^y A times B) for (y) without Logarithms

In this article, we will explore how to solve for (y) in the equation (x^y A times B) using straightforward mathematical methods. The solution may involve some repetitive steps, but it avoids the use of logarithms. This technique is particularly useful for those who do not have access to advanced computes or when using logarithms is not preferred or allowed.

Original Question

This article begins by addressing a question posed by Jobayed Shah: if xy A × B, how can one find the value of (y) when the values of (x), (A), and (B) are known? We will dive into the process of finding (y) using simple mathematics, making the solution accessible and understandable for all audiences.

Revised Question

Subsequently, we will explore how to solve the same problem without using logarithms. The key is to utilize a simple and repetitive method known as the "Guess My Number" technique, also called trial and error. This method is straightforward and can be applied in situations where advanced calculators or logarithmic functions are not available.

Mathematical Foundations

To begin, let's memorize two important inverse relationships:

1. Exponential Form: (b^{x} y)

When we have the equation in this form, we can easily find the exponent (x) if we know (b) and (y).

2. Logarithmic Form: (log_{b}y x)

This form is the inverse of the exponential form and allows us to find the exponent (x) when we know (b) and (y).

Solving the Given Equation

Given the equation (x^y A times B), we can manipulate it to:

(log_{x}(A times B) y)

However, since we want to avoid logarithms, we will use an iterative method, sometimes called the "Guess My Number" or trial and error technique.

Step-by-Step Solution

Let's assume the given values are:

(A 12) (B 2) (x 6)

Substituting these into the equation:

(6^y 12 times 2)

(6^y 24)

Step 1: Calculate the Target Value

First, we calculate (A times B):

(A times B 12 times 2 24)

Now, our equation is (6^y 24).

Step 2: Set Up Initial Bounds

We know that:

(6^1 6 (6^2 36 > 24)

Thus, the initial bounds are:

(L 1.000) (H 2.024) (where (H 2) and adding a small increment to avoid repeated rounding errors)

Step 3: Iterative Loop

In this step, we will keep updating our guess until the result is close enough to the target value. Here's the iterative loop:

Set (G), our initial guess, to the average of (L) and (H):

(G frac{L H}{2})

Calculate (6^G):

Compare the result with 24:

If the result is too low, set (L G) and try again. If the result is too high, set (H G) and try again. If the result is close enough, stop the loop and (G) is the solution for (y).

Let's perform a few iterations:

Iteration 1:

(G_1 frac{1 2.024}{2} 1.512)

(6^{1.512} approx 22.879)

Since 22.879

Iteration 2:

(G_2 frac{1.512 2.024}{2} 1.768)

(6^{1.768} approx 23.767)

Since 23.767

We can continue this process until the result is sufficiently close to 24. After a few more iterations, we will find our solution.

Conclusion

This method, while simple, can be quite effective for solving equations of the form (x^y A times B) without the need for logarithms. By starting with a range and iteratively refining our guess, we can find the value of (y) accurately.

Keywords: simple mathematics, solving for exponent, logarithm-free solution