Solving Exponential Equations: A Comprehensive Guide

Exponential equations can be challenging to solve, but with the right tools and techniques, you can find the solution efficiently. This article provides a detailed guide on how to solve the equation 0.88x frac{1}{2} using various methods, including logarithms and properties of exponents.

Introduction to Exponential Equations

In mathematics, an exponential equation is an equation involving an unknown variable in the exponent. Such equations are common in fields like finance, physics, and engineering. The structure of the exponential equation we are dealing with is 0.88x frac{1}{2}.

Method 1: Using Logarithms (Natural)

The first approach to solving the equation involves taking the natural logarithm of both sides:

Take the natural logarithm (ln) of both sides:

ln(0.88x) ln(frac{1}{2})

Apply the power rule of logarithms to bring the exponent down:

x cdot ln(0.88) ln(frac{1}{2})

Solve for x:

x frac{ln(frac{1}{2})}{ln(0.88)}

Calculate the values using a calculator:

Substitute these values into the equation:

x

Method 2: Using Logarithm Definition

The logarithm is the inverse of exponentiation, b^E X leftrightarrow log_bX E. Applying this principle to our equation:

Equate the logarithms to find x:

x log_{0.88}left(frac{1}{2}right) 5.4223...

Method 3: Using Base-10 Logarithms

Another approach is to use the base-10 logarithm (log10) of both sides:

Take the base-10 logarithm of both sides:

x log_{10}(0.88) log_{10}left(frac{1}{2}right)

Solve for x:

x frac{log_{10}(frac{1}{2})}{log_{10}(0.88)}

Calculate the values using a calculator or Python code:

from math import log10 print(-log10(2)/log10(0.88))

This will yield:

x

Method 4: Simplified Algebraic Approach

A fourth method involves directly manipulating the equation without logarithms. Begin by expressing the equation:

0.88x 1/2

0.88x 0.5

Take the common logarithm (log10) of both sides:

x log10(0.88) log10(0.5)

Solve for x:

x frac{log10(0.5)}{log10(0.88)}

Calculating this, you get:

x 0.5 / 0.88

x 0.56818

x 0.5682 (rounded to 4 decimal places)

Conclusion

Each of the methods mentioned above can be employed to solve 0.88x frac{1}{2}. The solutions derived from these methods confirm that x approx 5.43. Understanding and practicing these techniques will help you tackle more complex exponential equations in the future.