Logarithm Exercises: Get Answers & Solutions Here!

Dec 5, 2025 by Tom Lembong 51 views

Need Answers for These Logarithm Exercises?

Hey there, math enthusiasts! Having trouble cracking those logarithm problems? Don't sweat it; logarithms can be tricky, but with a bit of guidance, you'll be solving them like a pro in no time. This article breaks down the core concepts and provides step-by-step explanations to help you conquer those exercises. Let's dive in!

Understanding Logarithms: The Basics

Before we jump into specific exercises, let's make sure we're all on the same page about what logarithms actually are. At its heart, a logarithm answers the question: "What exponent do I need to raise a base to in order to get a certain number?"

Think of it like this: Exponents and logarithms are inverse operations, just like addition and subtraction, or multiplication and division. If 2 raised to the power of 3 (2³) equals 8, then the logarithm base 2 of 8 (log₂ 8) equals 3. See the connection? The base is the number being raised to a power, the exponent is the power itself, and the result is the number you're trying to reach.

The general form of a logarithm is: logₐ(x) = y, which translates to aʸ = x.

a is the base of the logarithm.
x is the argument of the logarithm (the number you're taking the logarithm of).
y is the exponent (the answer to the logarithmic equation).

Key Properties of Logarithms

Understanding these properties is crucial for solving logarithmic exercises. These rules will help you simplify complex expressions and manipulate equations to isolate the variable you're trying to find. Knowing these properties is like having a secret weapon in your math arsenal.

Product Rule: logₐ(xy) = logₐ(x) + logₐ(y). The logarithm of a product is equal to the sum of the logarithms of the individual factors. Example: log₂(4 * 8) = log₂(4) + log₂(8) = 2 + 3 = 5
Quotient Rule: logₐ(x/y) = logₐ(x) - logₐ(y). The logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator. Example: log₂(16 / 2) = log₂(16) - log₂(2) = 4 - 1 = 3
Power Rule: logₐ(xⁿ) = n * logₐ(x). The logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number. Example: log₂(8⁵) = 5 * log₂(8) = 5 * 3 = 15
Change of Base Rule: log_b(a) = log_c(a) / log_c(b). This rule allows you to change the base of a logarithm, which is especially useful when using a calculator that only has common (base 10) or natural (base e) logarithms. For example, if you need to find log₅(20) but your calculator only does base-10 logarithms, you can calculate it as log₁₀(20) / log₁₀(5).

Common Logarithms and Natural Logarithms

Common Logarithm: This is a logarithm with base 10, written as log₁₀(x) or simply log(x). It's the logarithm most calculators default to.
Natural Logarithm: This is a logarithm with base e (Euler's number, approximately 2.71828), written as ln(x). Natural logarithms are frequently used in calculus and other advanced mathematics.

Types of Logarithm Exercises and How to Solve Them

Logarithm exercises come in various forms, each requiring a slightly different approach. Let's explore some common types and strategies for tackling them. Remember to practice consistently, and don't be afraid to ask for help when you get stuck.

Evaluating Logarithms:

These exercises ask you to find the value of a logarithm. For example: log₂(32) = ?

Solution: You're asking, "What power do I need to raise 2 to in order to get 32?" Since 2⁵ = 32, then log₂(32) = 5.

Tips: If the argument is a power of the base, the answer is straightforward. If not, try to rewrite the argument as a power of the base. If you are allowed to use a calculator, use the change of base formula. log₂(32) = ln(32) / ln(2) = 5.
Solving Logarithmic Equations:

These exercises involve solving for an unknown variable within a logarithmic equation. For example: log₃(x) = 4

Solution: Convert the logarithmic equation to its exponential form: 3⁴ = x. Therefore, x = 81.

Tips: Isolate the logarithmic term first. Then, convert the equation to exponential form. Always check your solutions to make sure they don't result in taking the logarithm of a negative number or zero, as these are undefined.
Simplifying Logarithmic Expressions:

These exercises involve using the properties of logarithms to simplify complex expressions. For example: log₂(8x) - log₂(4)

Solution: Use the quotient rule and product rule to simplify: log₂(8x) - log₂(4) = log₂(8x/4) = log₂(2x) = log₂(2) + log₂(x) = 1 + log₂(x).

Tips: Identify which properties of logarithms can be applied to simplify the expression. Look for opportunities to combine or separate logarithmic terms.
Expanding Logarithmic Expressions:

These exercises involve using the properties of logarithms to expand a single logarithmic term into multiple terms. For example: log(x²y/z)

Solution: Use the product, quotient, and power rules to expand: log(x²y/z) = log(x²y) - log(z) = log(x²) + log(y) - log(z) = 2log(x) + log(y) - log(z).

Tips: Carefully apply the properties of logarithms, paying attention to signs and exponents.

Example Problems and Step-by-Step Solutions

Let's work through a few more examples to solidify your understanding. Follow along carefully, and try to solve them yourself first before looking at the solutions. That's the best way to learn!

Example 1: Solve for x: 2log₅(x) = log₅(9)

Solution:

Use the power rule to rewrite the left side: log₅(x²) = log₅(9)
Since the logarithms have the same base, we can equate the arguments: x² = 9
Solve for x: x = ±3
Check for extraneous solutions. Since we cannot take the logarithm of a negative number, x = -3 is not a valid solution. Therefore, x = 3.

Example 2: Simplify: ln(e⁵) + log₂(1/4)

Solution:

Simplify ln(e⁵). Since the natural logarithm has a base of e, ln(e⁵) = 5.
Simplify log₂(1/4). We're asking, "What power do I need to raise 2 to in order to get 1/4?" Since 2⁻² = 1/4, then log₂(1/4) = -2.
Combine the results: 5 + (-2) = 3.

Example 3: Expand: log₃((x²y)/√z)

Solution:

Rewrite the square root as an exponent: log₃((x²y)/z^(1/2))
Use the quotient rule: log₃(x²y) - log₃(z^(1/2))
Use the product rule: log₃(x²) + log₃(y) - log₃(z^(1/2))
Use the power rule: 2log₃(x) + log₃(y) - (1/2)log₃(z)

Tips and Tricks for Mastering Logarithms

Practice Regularly: The more you practice, the more comfortable you'll become with logarithms.
Know Your Properties: Memorize and understand the properties of logarithms. They are your best tools for solving exercises.
Check Your Solutions: Always check your solutions, especially when solving logarithmic equations, to avoid extraneous solutions.
Break Down Complex Problems: Divide complex problems into smaller, more manageable steps.
Use Online Resources: There are many online resources available to help you learn about logarithms, including tutorials, practice problems, and calculators.
Don't Be Afraid to Ask for Help: If you're struggling with logarithms, don't hesitate to ask your teacher, a tutor, or a classmate for help.

Conclusion

Logarithms might seem daunting at first, but with a solid understanding of the basics and consistent practice, you can master them. Remember the key properties, practice different types of exercises, and don't be afraid to seek help when needed. Keep practicing, and you'll be amazed at how quickly you improve! So go ahead, tackle those logarithm exercises with confidence, guys! You got this! Good luck, and happy problem-solving!