Solving Logarithm Problems Using Change-of-Base Formula and Laws of Logarithms

Logarithms are a fundamental concept in mathematics, often encountered in various applications such as acoustics, complex analysis, and higher-level mathematics. This article will explore how to solve logarithmic problems using the change-of-base formula and the laws of logarithms, providing detailed solutions to specific problems.

Understanding the Change-of-Base Formula

The change-of-base formula is a powerful tool that allows us to solve logarithms of any base in terms of a common or preferred base. The formula is given by:

log_mx lognx}{lognm}

This formula can greatly simplify solving logarithms, especially when the base is not a common base like 10 or e. Let's walk through the steps to apply the formula in solving a logarithmic problem.

Problem Statement

Consider a problem where we need to solve for loga}{b}x. We will use the change-of-base formula and the given base to simplify and solve the problem.

Step-by-Step Solution

Given: loga}{b}x

Using the change-of-base formula, let’s set m and n a. The formula becomes:

logx logax}{loga}

Further decomposing the denominator:

logaaa - logab 1 - logab

This simplifies the expression to:

logx logax}{1 - logab}

Now, substituting this into the original equation:

logax}{logx} logax}{logax}{1 - logab} logax · ab}{logax}

Simplifying, we get:

1 - logab

Further Applications of Logarithm Laws

Now let's apply the laws of logarithms to solve two specific problems.

Problem 1: Solving logx81 4

Using the laws of logarithms, we have:

81 x^4

x 481

x 3

Note that x must be positive and real.

Problem 2: Solving log2x - 6 3

Using the laws of logarithms, we have:

x - 6 2^3

x - 6 8

x 14

Conclusion

In this article, we have discussed how to apply the change-of-base formula and laws of logarithms to solve specific logarithmic problems. By using these fundamental techniques, we can simplify and solve a variety of logarithmic equations.