Logarithmic Equations and Expressions

Hello! Welcome to the next lesson in our CAT preparation course.

In our previous session, we mastered the formulas for special series, like the sum of the first $n$ squares and cubes. As planned, we're now shifting from series to another cornerstone of algebra: logarithms.

Today's lesson is designed to help you solve equations and simplify expressions using the properties of logarithms. This is a topic that frequently appears in the CAT exam, often in combination with other algebraic concepts. For someone with a Computer Science background, the idea of a logarithm as the inverse of an exponential function might feel familiar—it's the same principle behind why an efficient search algorithm has a time complexity of $O (lo g n)$ instead of $O (n)$ .

We'll start from the ground up, defining what a logarithm is, then move on to its powerful properties, and finally, apply these properties to solve CAT-level problems.

1. What is a Logarithm?

At its core, a logarithm is simply a way of asking a question: "What exponent do I need to put on this base to get that number?"

The formal definition is:
If $b^{x} = a$ , then we can write this in logarithmic form as $lo g_{b} a = x$ .

Here:

$b$ is the base.
$a$ is the argument.
$x$ is the logarithm (which is just the exponent).

For example, since $2^{5} = 32$ , we can write this as $lo g_{2} 32 = 5$ .

To get a solid grasp of this fundamental relationship and the essential conditions for a logarithm to be defined, let's watch the beginning of a video from Rodha, a channel focused on CAT preparation.

| Logarithms part 1 |Algebra| CAT Preparation |

This video explains the core definition of a logarithm and the important constraints on its base and argument.

Watch the segment from 00:49 to 03:13. Focus on understanding the equivalence between the exponential form (a^x = n) and the logarithmic form (x = log_a n). Also, pay close attention to the three conditions for a logarithm to be defined.

To summarize the crucial constraints you just saw: for $lo g_{b} a$ to be defined:

The base $b$ must be positive: $b > 0$ .
The base $b$ cannot be 1: $b \neq = 1$ .
The argument $a$ must be positive: $a > 0$ .

These rules are not just theoretical; they are critical when solving equations, as they help you eliminate invalid or "extraneous" solutions.

2. The Laws of Logarithms

The real power of logarithms comes from their properties, which allow us to transform complex calculations. These rules are the heart of this topic.

Here are the main laws you need to master:

Law Name	Formula	What it does
Product Rule	$lo g_{b} (mn) = lo g_{b} m + lo g_{b} n$	Turns multiplication into addition.
Quotient Rule	$lo g_{b} (\frac{m}{n}) = lo g_{b} m - lo g_{b} n$	Turns division into subtraction.
Power Rule	$lo g_{b} (m^{p}) = p \cdot lo g_{b} m$	Brings an exponent down as a multiplier.
Change of Base	$lo g_{b} a = \frac{l o g _{c} a}{l o g _{c} b}$	Lets you change to any new base $c$ .

From these, we can also derive other incredibly useful properties:

Base & Argument Power Rule: $lo g_{b^{q}} a^{p} = \frac{p}{q} lo g_{b} a$
Base Switch Rule: $lo g_{b} a = \frac{1}{l o g _{a} b}$
Identity Rules: $lo g_{b} b = 1$ and $lo g_{b} 1 = 0$
Interchange Rule: $a^{l o g_{b} c} = c^{l o g_{b} a}$

To see how these rules are applied in practice, let's return to the Rodha video. The instructor provides clear, CAT-relevant examples for each property.

| Logarithms part 1 |Algebra| CAT Preparation |

This segment covers the essential properties of logarithms with practical examples.

Watch the video from 03:13 to 17:11. It's a longer segment, so feel free to pause as you go. Focus on: The Change of Base formula (starts at 03:13). The Product, Quotient, and Power rules, including the very useful rule for powers on both the base and argument (starts at 09:55). The Interchange rule (a^(log_b c) = c^(log_b a)), which is a handy shortcut (starts at 14:48).

Test your understanding!

Simplify the expression: $lo g_{2} 16 + lo g_{3} (\frac{1}{9}) - lo g_{25} 5$

Show answer

Let's break it down term by term:

$lo g_{2} 16$ : We ask, "2 to what power is 16?" Since $2^{4} = 16$ , this is 4.
$lo g_{3} (\frac{1}{9})$ : We ask, "3 to what power is 1/9?" Since $9 = 3^{2}$ , we have $1/9 = 1/ 3^{2} = 3^{- 2}$ . So, this is -2.
$lo g_{25} 5$ : We ask, "25 to what power is 5?" Since $25 = 5$ , we have $2 5^{1/2} = 5$ . So, this is 1/2.

Combining them: $4 + (- 2) - \frac{1}{2} = 2 - 0.5 = 1.5$ .

3. Solving Logarithmic Equations

Now, let's apply these properties to achieve our main goal: solving equations. The general strategy is:

Condense: Use the log properties to combine multiple log terms into a single logarithm on one or both sides of the equation.
Solve:
- If you have $lo g_{b} A = lo g_{b} C$ , you can set $A = C$ .
- If you have $lo g_{b} A = C$ , convert it to exponential form: $b^{C} = A$ .
Check: This is a critical final step. Substitute your solutions back into the original equation to ensure that all logarithm arguments are positive. Any solution that results in the log of a non-positive number must be discarded.

The next video segment will walk you through a classic CAT-style problem, highlighting the importance of checking for extraneous roots.

| Logarithms part 1 |Algebra| CAT Preparation |

Watch how the instructor solves a logarithmic equation. Pay very close attention to the end, where he checks the validity of the solutions obtained.

Watch the segment from 17:29 to 22:36. Follow the steps: Making the bases the same. Using the power rule to manipulate the equation. Solving the resulting quadratic equation. Checking both solutions against the original logarithmic expressions.

The process of checking for extraneous solutions, as shown in the video, cannot be skipped. Here is a clear, written-out example that reinforces this process.

Solving Logarithmic Equations with Extraneous Solutions — This image provides a step-by-step solution for the equation $\log_4(1-3x) + \log_4(3-x) = 2$. It demonstrates using the product rule to combine the logs, converting to exponential form to get a quadratic equation, and finally, checking both potential solutions ($x=-1$ and $x=13/3$). Notice how $x=13/3$ is rejected because it makes the arguments of the original logs negative.

Test your understanding!

Solve for $x$ : $lo g_{6} (x + 5) + lo g_{6} x = 2$

Show answer

Condense: Using the product rule, we combine the logs on the left side:
$lo g_{6} ((x + 5) x) = 2$
$lo g_{6} (x^{2} + 5 x) = 2$
Solve: Convert to exponential form:
$x^{2} + 5 x = 6^{2}$
$x^{2} + 5 x = 36$
$x^{2} + 5 x - 36 = 0$
This is a quadratic equation. We can factor it by finding two numbers that multiply to -36 and add to 5. These are 9 and -4.
$(x + 9) (x - 4) = 0$
The potential solutions are $x = - 9$ and $x = 4$ .
Check:
- Check $x = - 9$ : If we plug this into the original equation, the first term becomes $lo g_{6} (- 9 + 5) = lo g_{6} (- 4)$ . The argument is negative, so this is not a valid solution.
- Check $x = 4$ : The terms become $lo g_{6} (4 + 5) = lo g_{6} (9)$ and $lo g_{6} (4)$ . Both arguments are positive. This is a valid solution.

Therefore, the only solution is $x = 4$ .

Conclusion

Today we've unpacked logarithms, a topic that often seems intimidating but is perfectly manageable once you understand its rules. We started with the basic definition—that a logarithm is an exponent—and then systematically worked through the properties that allow us to manipulate and simplify complex expressions.

Key Takeaways:

A logarithm is the inverse of an exponent: $lo g_{b} a = x$ is the same as $b^{x} = a$ .
The core properties (Product, Quotient, Power rules) allow you to convert between multiplication/division and addition/subtraction.
The Change of Base rule is essential for making terms compatible.
When solving logarithmic equations, the non-negotiable final step is to check your solutions in the original equation to ensure all log arguments are positive.

Preview of the Next Lesson:
In our next lesson, we will continue our exploration of key algebraic functions by looking at modulus and greatest integer functions. These functions have unique properties and frequently appear in CAT questions, requiring special techniques to solve equations and inequalities involving them.

Recommended resources

1. What is a Logarithm?

2. The Laws of Logarithms

3. Solving Logarithmic Equations

Conclusion