Simplifying Expressions with Logarithms and Exponentials

From Exponents to Logarithms

The properties of logarithms are not arbitrary; they are a direct consequence of the rules for exponents. Let's begin by ensuring those rules are fresh in your mind.

An expression like $b^x$ involves a base $b$ raised to an exponent $x$. The fundamental rules govern how we handle multiplication, division, and nested powers.

To summarize, for any valid base $b$ and real exponents $x$ and $y$:

• Product Rule: $b^x\cdot b^y=b^{x+y}$
• Quotient Rule: $\frac{b^x}{b^y}=b^{x-y}$
• Power Rule: $(b^x{)}^y=b^{xy}$

These three rules are the seeds from which all the properties of logarithms grow.

Now, what is a logarithm? It is simply the inverse of an exponential function. It answers the question: "To what exponent must we raise a base to obtain a certain value?"

This inverse relationship, $b^y=x\iff {\log }_b(x)=y$, holds under specific conditions that we can define using set theory from our last lesson:

• The base $b$ must be a positive real number not equal to 1, i.e., $b\in (0,1)\cup (1,\infty )$.
• The argument $x$ must be a positive real number, i.e., $x\in (0,\infty )$. The set of positive real numbers is the domain of the logarithm function.
• The result $y$ can be any real number, so the range is $(-\infty ,\infty )$.

Two bases are particularly important:

• Common Logarithm: Base 10, written as $\log (x)$.
• Natural Logarithm: Base $e$, written as $\ln (x)$. The number $e\approx 2.71828$ is a fundamental mathematical constant, like $\pi$, that arises naturally in calculus and growth processes. We will explore it more in later modules.

Deriving the Properties of Logarithms

Because you are interested in a rigorous foundation, it is not enough to just state the properties. Understanding where they come from is key. Each property of logarithms is a direct translation of an exponent rule, seen through the lens of the inverse relationship.

The following video provides excellent, step-by-step proofs for the three core laws of logarithms. As you watch, notice how each proof follows a similar pattern:

1. Define the logarithmic terms as variables.
2. Convert the logarithmic equations to their exponential equivalents.
3. Apply the corresponding exponent rule (product, quotient, or power).
4. Convert the result back into a logarithmic equation.

These proofs beautifully illustrate the connection we've been discussing. The properties aren't new rules to memorize, but rather the exponent rules you already know, just expressed in a different language.

The image below provides a complete summary of these properties, including a few others that derive from the definition. It's a useful reference sheet.

The Change of Base Formula

What if you need to calculate a logarithm with a base that isn't on your calculator, like ${\log }_4(5)$? The change of base formula allows you to convert any logarithm into a ratio of logarithms with a different, more convenient base (like 10 or $e$).

The formula is: ${\log }_a(x)=\frac{{\log }_b(x)}{{\log }_b(a)}$

For a clear derivation of this essential formula, please consult the following resource.

This formula shows that all logarithmic functions are simply scaled versions of one another. The graph of ${\log }_4(x)$ has the same shape as $\ln (x)$; it's just stretched or compressed vertically by a factor of $1/\ln (4)$.

Applying the Properties: Expanding and Condensing

Now that we understand the properties and their origins, let's put them to use. Simplification of logarithmic expressions generally takes two forms:

1. Expanding: Taking a single, complex logarithm and breaking it down into a sum or difference of simpler logarithms.
2. Condensing: Taking a series of simple logarithms and combining them into a single, more compact logarithmic expression.

These skills are crucial for solving logarithmic equations and for manipulating expressions in calculus and beyond.

A Critical Subtlety: Mind the Domain

As you become proficient, it's vital to be aware of a subtlety that can trip up even experienced students. Applying the rules of logarithms can sometimes alter the domain of the expression. This is a point of rigor that is often overlooked but is critical for formal mathematics.

For instance, consider the functions $f(x)=\log (x^2)$ and $g(x)=2\log (x)$.

• The domain of $f(x)$ is all real numbers except 0, since $x^2$ is positive for any non-zero $x$.
• The domain of $g(x)$ is only positive real numbers ($x>0$).

Because their domains are different, $f(x)$ and $g(x)$ are not identical functions. The identity $\log (x^2)=2\log (x)$ is only strictly true for $x>0$. The fully correct identity for all $x\ne 0$ is $\log (x^2)=2\log |x|$.

Conclusion

In this lesson, we have revisited exponents and used their properties as a foundation to build a robust understanding of logarithms. You've learned not only what the properties of logarithms are but also why they are true, by examining their formal proofs. We then practiced applying these properties to expand and condense expressions, a key algebraic skill.

Key Takeaways:

• Logarithms are the inverse of exponential functions.
• The properties of logarithms (Product, Quotient, and Power Rules) are direct consequences of the corresponding exponent rules.
• Logarithmic expressions can be expanded into simpler terms or condensed into a single term.
• The Change of Base formula allows any logarithm to be expressed in terms of any other base, showing that all logarithmic functions are related by a simple scaling factor.
• Applying logarithmic properties can change the domain of an expression, which requires careful attention for mathematically precise work.

In our next lesson, we will introduce summation and product notation ($\Sigma$ and $\Pi$). These are powerful tools for representing and manipulating the long sums and products that frequently appear in machine learning algorithms, such as in the definition of a neuron's activation or a model's cost function.