Express The Quantity As A Single Logarithm

Introduction

In algebraand calculus, it is often useful to express the quantity as a single logarithm rather than as a sum, difference, or multiple of several logarithmic terms. Doing so simplifies expressions, makes solving equations more straightforward, and reveals the underlying structure of exponential relationships. Whether you are simplifying a complex logarithmic expression before differentiation, condensing terms in a proof, or preparing a formula for numerical evaluation, the ability to combine logarithms into one compact form is a fundamental skill. This article walks you through the theory, the step‑by‑step procedure, practical examples, and common pitfalls so you can confidently rewrite any logarithmic quantity as a single log.

Detailed Explanation

A logarithm answers the question: “To what exponent must the base be raised to obtain a given number?” Symbolically, for a positive base (b\neq1) and a positive argument (x),

[ \log_b x = y \quad\Longleftrightarrow\quad b^{,y}=x . ]

Because logarithms are the inverse of exponentiation, they inherit three core properties that mirror the rules for powers:

1. Product Rule – (\displaystyle \log_b (MN)=\log_b M+\log_b N)
2. Quotient Rule – (\displaystyle \log_b \left(\frac{M}{N}\right)=\log_b M-\log_b N)
3. Power Rule – (\displaystyle \log_b (M^{k})=k,\log_b M)

These identities allow us to move back and forth between a single logarithm and a combination of several logarithms. When we are asked to express the quantity as a single logarithm, we start with an expression that may contain added, subtracted, or multiplied log terms (including coefficients) and apply the rules in reverse: we combine sums into products, differences into quotients, and coefficients into exponents inside the log.

It is essential to keep the domain in mind: the argument of any logarithm must be positive. Whenever we combine terms, we must verify that the resulting argument remains positive for the values of the variables under consideration; otherwise the combined expression is not equivalent to the original.

Step‑by‑Step Concept Breakdown

Below is a systematic procedure you can follow for any logarithmic expression that needs to be condensed into a single log.

1. Identify the Base

All logarithms in the expression must share the same base. If they differ, use the change‑of‑base formula

[ \log_b x = \frac{\log_k x}{\log_k b} ]

to rewrite every term with a common base (often base 10 or (e)).

2. Deal with Coefficients (Power Rule in Reverse)

Any coefficient (c) in front of a log can be moved inside as an exponent:

[ c\log_b M = \log_b (M^{c}) . ]

If the coefficient is a fraction, treat it as a rational exponent (e.g., (\frac12\log_b M = \log_b \sqrt{M})).

3. Combine Added Logs (Product Rule)

Replace each sum of logs with the log of a product:

[ \log_b M + \log_b N = \log_b (MN). ]

If more than two terms are added, apply the rule repeatedly or combine all at once: [ \log_b M_1 + \log_b M_2 + \dots + \log_b M_k = \log_b (M_1M_2\cdots M_k). ]

4. Combine Subtracted Logs (Quotient Rule)

Replace each difference of logs with the log of a quotient:

[ \log_b M - \log_b N = \log_b \left(\frac{M}{N}\right). ]

When a sum and a difference appear together, handle the additions first (product) then the subtractions (quotient), or vice‑versa—just keep track of numerator and denominator groupings.

5. Simplify the Argument

After applying the product and quotient rules, you will have a single log whose argument may contain powers, products, or quotients. Simplify that algebraic expression as much as possible (factor, cancel, reduce fractions).

6. Verify Domain

State any restrictions on the variables that guarantee the argument of the final log is positive. If the original expression had separate domain restrictions (e.g., each individual log required its argument > 0), the combined expression must respect the intersection of those restrictions.

Following these six steps guarantees that you have correctly expressed the quantity as a single logarithm.

Real Examples

Example 1 – Basic Combination

Express (\displaystyle 3\log_2 x - \log_2 (x+4) + \frac12\log_2 (x-1)) as a single logarithm.

Solution

1. All logs already have base 2.
2. Apply the power rule:
   [ 3\log_2 x = \log_2 (x^{3}),\qquad \frac12\log_2 (x-1)=\log_2 \big((x-1)^{1/2}\big)=\log_2 \sqrt{x-1}. ]
3. Combine the added logs (product):
   [ \log_2 (x^{3}) + \log_2 \sqrt{x-1}= \log_2 \big(x^{3}\sqrt{x-1}\big). ] 4. Apply the quotient rule for the subtraction:
   [ \log_2 \big(x^{3}\sqrt{x-1}\big) - \log_2 (x+4)=\log_2 \left(\frac{x^{3}\sqrt{x-1}}{x+4}\right). ]
4. The final expression is
   [ \boxed{\displaystyle \log_2 \left(\frac{x^{3}\sqrt{x-1}}{x+4}\right)} , ]
   with domain (x>1) (to keep (x-1>0) and the denominator non‑zero).

Example 2 – Different Bases

Express (\displaystyle \log_5 25 + 2\log_{10} 3 - \log_{10} 9) as a single logarithm (choose base 10 for the final answer).

Solution

1. Convert the base‑5 term to base 10: [ \log_5 2