Understanding and Simplifying Logarithmic Expressions Using the Change of Base Formula

Logarithms are powerful tools in mathematics, used widely in various fields including computer science, physics, and engineering. One common task involves simplifying logarithmic expressions that may appear complex at first glance. This article will walk you through the process of simplifying the expression (log_2 3 cdot log_3 4 cdot log_4 5 cdot log_5 6 cdot log_6 7 cdot log_7 8) using the change of base formula.

The Change of Base Formula

The change of base formula for logarithms is a fundamental tool that allows you to express a logarithm with any base as a quotient of logarithms with a common base. The formula is given by:

(log_a b frac{log_c b}{log_c a})

Where (c) is any positive number, usually 10 (common logarithm) or (e) (natural logarithm). The choice of base does not affect the final result, as the base in the numerator and denominator will cancel out.

Simplifying the Expression

Let's start with the given expression:

(log_2 3 cdot log_3 4 cdot log_4 5 cdot log_5 6 cdot log_6 7 cdot log_7 8)

Using the change of base formula, we rewrite each logarithm:

(log_2 3 frac{log 3}{log 2}) (log_3 4 frac{log 4}{log 3}) (log_4 5 frac{log 5}{log 4}) (log_5 6 frac{log 6}{log 5}) (log_6 7 frac{log 7}{log 6}) (log_7 8 frac{log 8}{log 7})

Substituting these into the original expression, we get:

(left(frac{log 3}{log 2}right) cdot left(frac{log 4}{log 3}right) cdot left(frac{log 5}{log 4}right) cdot left(frac{log 6}{log 5}right) cdot left(frac{log 7}{log 6}right) cdot left(frac{log 8}{log 7}right))

Notice that this product telescopes, meaning that intermediate terms cancel out:

(log 3) in the numerator of the first term cancels with (log 3) in the denominator of the second term. (log 4) in the numerator of the second term cancels with (log 4) in the denominator of the third term. (log 5), (log 6), and (log 7) cancel similarly.

After all cancellations, we are left with:

(frac{log 8}{log 2})

Further Simplification

Since (8 2^3), we can simplify (log 8) as follows:

(log 8 log 2^3 3 log 2)

Substituting this back into our expression, we get:

(frac{3 log 2}{log 2} 3)

Hence, the value of the original expression is:

(boxed{3})

Alternative Approach Using Natural Logarithms

An alternative approach is to use natural logarithms (ln) for simplification:

(log_2 3 frac{ln 3}{ln 2})

(log_3 4 frac{ln 4}{ln 3})

(log_4 5 frac{ln 5}{ln 4})

(log_5 6 frac{ln 6}{ln 5})

(log_6 7 frac{ln 7}{ln 6})

(log_7 8 frac{ln 8}{ln 7})

Multiplying these together:

(frac{ln 3}{ln 2} cdot frac{ln 4}{ln 3} cdot frac{ln 5}{ln 4} cdot frac{ln 6}{ln 5} cdot frac{ln 7}{ln 6} cdot frac{ln 8}{ln 7})

This simplifies to:

(frac{ln 8}{ln 2})

Since (ln 8 ln (2^3) 3 ln 2) the expression further simplifies to:

(frac{3ln 2}{ln 2} 3)

Thus, the value of the original expression is:

(boxed{3})

Summary

This article has demonstrated how to simplify logarithmic expressions using the change of base formula. By leveraging telescoping properties and basic logarithmic identities, we can solve seemingly complex problems with relative ease. Understanding these techniques is crucial for advanced mathematical analysis and problem solving.

Key Takeaways

The change of base formula allows you to convert logarithms into a quotient form. Telescoping cancels intermediate terms, leaving only the first and last terms. Natural logarithms (ln) can also be used to simplify the expression effectively.

Mastering these concepts will help you tackle more complex logarithmic expressions and enhance your problem-solving skills in mathematics and related fields.