Logarithm Rules Practice Problems - Master Log Laws

Practice logarithmic laws with step-by-step solutions. Master product, quotient, power rules, change of base formula, and natural logarithms through targeted exercises.

📚Master Logarithmic Laws Through Practice
  • Apply product rule: log_a(MN) = log_a(M) + log_a(N) to solve complex problems
  • Use quotient rule: log_a(M/N) = log_a(M) - log_a(N) for division inside logs
  • Master power rule: log_a(M^n) = n·log_a(M) to simplify exponential expressions
  • Convert between bases using change of base formula: log_b(x) = log_c(x)/log_c(b)
  • Solve logarithmic equations by converting to exponential form
  • Work with natural logarithms (ln) and understand e as the base

Understanding Rules of Logarithms

Complete explanation with examples

What Are Logarithmic Laws?

There are a few logarithmic laws worth knowing to make solving problems easier. The following laws are the main rules you will use. It should be noted that the letters a, m, n must be positive real numbers for these laws to be valid.

Logarithmic Laws

Constant Values:

It can be automatically determined that:

  • loga(1)=0 log_a\left(1\right)=0
  • loga(a)=1 log_a\left(a\right)=1

Basic Arithmetic Operations

Multiplication, division, subtraction, and addition operations between logarithms:

  • logaMN=logaM+logaN log_aMN=log_aM+log_aN
  • logaM/N=logaMlogaN log_aM/N=log_aM-log_aN
  • Loga(M)×Logn(D)=Logn(M)×Loga(D) Log_a\left(M\right)\times Log_n\left(D\right)=Log_n\left(M\right)\times Log_a\left(D\right)
  • LogaMn=nLogaM Log_aM^n=nLog_aM

Visual explanation of logarithmic rules showing log(x·y) equals log(x) plus log(y), and log(x/y) equals log(x) minus log(y), with arrows connecting each part for clarity.

Changing the Base of a Logarithm:

  • logb(x)=logc(x)/logc(b) log_b\left(x\right)=log_c\left(x\right)/log_c\left(b\right)
  • logb(c)=1/logc(b) log_b\left(c\right)=1/log_c\left(b\right)

Logarithmic change of base formula illustrated: log base b of a equals log base x of a divided by log base x of b, with arrows showing transformation from original form.

Derivative of the Logarithm:

fx=logb(x)fx=1/xln(b) fx=log_b\left(x\right)⇒f^{\prime}x=1/xln(b)

Integral of the Logarithm:

logb(x)dx=x×logb(x)1/ln(b)+C ∫log_b\left(x\right)dx=x\times log_b\left(x\right)-1/ln\left(b\right)+C

Detailed explanation

Practice Rules of Logarithms

Test your knowledge with 38 quizzes

\( 2\log_82+\log_83= \)

Examples with solutions for Rules of Logarithms

Step-by-step solutions included
Exercise #1

3log76= 3\log_76=

Step-by-Step Solution

To simplify the expression 3log76 3\log_76 , we apply the power property of logarithms, which states:

alogbc=logb(ca) a\log_b c = \log_b(c^a)

Step 1: Identify the given expression: 3log76 3\log_76 .

Step 2: Apply the power property of logarithms:

3log76=log7(63) 3\log_76 = \log_7(6^3)

Step 3: Calculate 63 6^3 :

63=6×6×6=36×6=216 6^3 = 6 \times 6 \times 6 = 36 \times 6 = 216

Step 4: Substitute back into the logarithmic expression:

log7(63)=log7216 \log_7(6^3) = \log_7216

Therefore, the simplified expression is log7216\log_7216.

Comparing with the answer choices, the correct choice is:

log7216 \log_7216

Answer:

log7216 \log_7216

Video Solution
Exercise #2

log75log72= \log_75-\log_72=

Step-by-Step Solution

To solve the problem, let's use the rules of logarithms:

  • Step 1: Recognize that we are dealing with the subtraction of logarithms sharing the same base, which calls for the identity logbMlogbN=logb(MN)\log_b M - \log_b N = \log_b \left(\frac{M}{N}\right).
  • Step 2: Apply this identity to the expression log75log72\log_7 5 - \log_7 2.
  • Step 3: Realize that this can thus be expressed as a single logarithm: log7(52)\log_7 \left(\frac{5}{2}\right).
  • Step 4: Simplify the fraction, yielding log72.5\log_7 2.5.

Therefore, the simplification results in the expression: log72.5\log_7 2.5.

This matches the correct answer from the given choices.

Answer:

log72.5 \log_72.5

Video Solution
Exercise #3

log29log23= \log_29-\log_23=

Step-by-Step Solution

To solve the problem of evaluating log29log23\log_2 9 - \log_2 3, we apply the properties of logarithms as follows:

  • Step 1: Recognize that the expression uses a subtraction of logarithms with the same base: log29log23\log_2 9 - \log_2 3.
  • Step 2: Use the logarithmic subtraction rule: logbAlogbB=logb(AB)\log_b A - \log_b B = \log_b \left(\frac{A}{B}\right).
  • Step 3: Simplify using this rule: log29log23=log2(93)\log_2 9 - \log_2 3 = \log_2 \left(\frac{9}{3}\right).
  • Step 4: Perform the division: 93=3\frac{9}{3} = 3.
  • Step 5: Therefore, log2(93)=log23\log_2 \left(\frac{9}{3}\right) = \log_2 3.

Thus, the simplified and evaluated result is log23 \log_2 3 .

Answer:

log23 \log_23

Video Solution
Exercise #4

logmn×logzr= \log_mn\times\log_zr=

Step-by-Step Solution

To solve the problem of finding what logmn×logzr \log_m n \times \log_z r equals, we will apply some rules of logarithms:

1. Restate the problem: We need to determine the expression that logmn×logzr \log_m n \times \log_z r is equivalent to. 2. Key information: We have two logarithms: logmn \log_m n and logzr \log_z r . 3. Potential approaches: Use the change of base formula for logarithms. 4. Key formulas: The change of base formula for logarithms states logab=logcblogca \log_a b = \frac{\log_c b}{\log_c a} . 5. Chosen approach: Use the change of base to express each log\log in terms of a common base and simplify. 6. Outline steps: - Apply the change of base formula to each logarithmic term. - Simplify the expression. 7. Assumptions: Assume variables m,n,z,r m, n, z, r are positive real numbers and bases (m m and z z ) are not equal to 1. 8. Simplification: Change each logarithm to a form using a common base logarithm for easier simplification. 11. Multiple choice: We will check which answer choice represents the derived expression. 12. Common mistakes: Forgetting to apply the change of base properly or incorrect simplification.

Let's work through the solution step-by-step:

  • Step 1: Apply the change of base formula.
  • Step 2: Simplify the expression using properties of logarithms.
  • Step 3: Identify the expression among the given choices.

Now, let's apply the steps:

Step 1: Use the change of base formula.
By the change of base formula, we know that:

logmn=logknlogkm \log_m n = \frac{\log_k n}{\log_k m}
logzr=logkrlogkz \log_z r = \frac{\log_k r}{\log_k z}

for any base k k . Using the natural logarithm base (ln) (\ln) for simplicity, we substitute into these expressions:

logmn=lnnlnm \log_m n = \frac{\ln n}{\ln m}
logzr=lnrlnz \log_z r = \frac{\ln r}{\ln z}

Step 2: Simplify.

Now, multiply the two expressions:

logmn×logzr=(lnnlnm)×(lnrlnz) \log_m n \times \log_z r = \left(\frac{\ln n}{\ln m}\right) \times \left(\frac{\ln r}{\ln z}\right)

Simplifying, we get:

=lnn×lnrlnm×lnz = \frac{\ln n \times \ln r}{\ln m \times \ln z}

Step 3: Expression equivalence analysis.

By rearranging the terms using logarithmic properties, it follows that the expression simplifies to:

logzn×logmr \log_z n \times \log_m r

Therefore, the solution to the problem is logzn×logmr \log_z n \times \log_m r .

This matches option 1 in the multiple choice answers provided.

Answer:

logzn×logmr \log_zn\times\log_mr

Video Solution
Exercise #5

log49×log137= \log_49\times\log_{13}7=

Step-by-Step Solution

To solve the problem log49×log137 \log_49 \times \log_{13}7 , we'll employ the change of base formula for logarithms:

  • Step 1: Apply the change of base formula to each logarithm.
  • Step 2: Use logarithm properties and analyze transformations for a match with choices.

Now, let's work through each step:
Step 1: Use the change of base formula on each log:
log49=loga9loga4 \log_49 = \frac{\log_a 9}{\log_a 4} and log137=logb7logb13 \log_{13}7 = \frac{\log_b 7}{\log_b 13} , where a a and b b are arbitrary positive bases.
Both expressions use a common base not relevant for the solution but illustrate the transformation ability.

Step 2: We'll recombine and look for products that can utilize these, such as:

log139×log47 \log_{13}9\times\log_47 becomes loga9loga13×logb7logb4 \frac{\log_a 9}{\log_a 13} \times \frac{\log_b 7}{\log_b 4}
Applying cross multiplication or iteration forms, the structure aligns with the multiplication identity for this problem due to independence of base.

Therefore, the transformed expression satisfying the criteria is log139×log47 \log_{13}9\times\log_47 .

Answer:

log139×log47 \log_{13}9\times\log_47

Video Solution

Frequently Asked Questions

What are the basic logarithm rules I need to memorize?

+
The essential logarithm rules are: log_a(1) = 0, log_a(a) = 1, log_a(MN) = log_a(M) + log_a(N), log_a(M/N) = log_a(M) - log_a(N), and log_a(M^n) = n·log_a(M). These five rules form the foundation for solving most logarithmic problems.

How do I solve logarithmic equations step by step?

+
To solve logarithmic equations: 1) Use logarithm rules to simplify both sides, 2) Get a single logarithm on each side if possible, 3) Convert to exponential form (if log_a(x) = y, then a^y = x), 4) Solve the resulting equation, 5) Check your answer in the original equation to ensure it's in the domain.

What is the change of base formula and when do I use it?

+
The change of base formula is log_b(x) = log_c(x)/log_c(b), where c can be any positive base (usually 10 or e). Use this formula when you need to calculate a logarithm with a base that's not available on your calculator, or when solving equations with different bases.

What's the difference between common logarithms and natural logarithms?

+
Common logarithms have base 10 (written as log or log₁₀) while natural logarithms have base e ≈ 2.718 (written as ln). Natural logarithms are used extensively in calculus and exponential growth/decay problems, while common logarithms are often used in scientific applications and pH calculations.

Why can't I take the logarithm of negative numbers?

+
Logarithms of negative numbers are undefined in real numbers because no real power of a positive base can equal a negative number. For example, there's no real number x where 10^x = -5. The domain of logarithmic functions includes only positive real numbers.

How do I expand log expressions using logarithm rules?

+
To expand logarithms: 1) Use log_a(MN) = log_a(M) + log_a(N) for products, 2) Use log_a(M/N) = log_a(M) - log_a(N) for quotients, 3) Use log_a(M^n) = n·log_a(M) for powers. For example, log₃(x²y/z) = 2log₃(x) + log₃(y) - log₃(z).

What are the most common mistakes when working with logarithms?

+
Common logarithm mistakes include: forgetting domain restrictions (arguments must be positive), incorrectly applying rules like log(a+b) ≠ log(a) + log(b), mixing up product and quotient rules, and failing to check solutions in the original equation. Always verify your answers are in the proper domain.

How do logarithms relate to exponential functions?

+
Logarithms and exponentials are inverse functions. If y = log_a(x), then a^y = x. This relationship is crucial for solving logarithmic equations by converting them to exponential form. For example, if log₂(x) = 3, then 2³ = x, so x = 8.

More Rules of Logarithms Questions

Practice by Question Type

More Resources and Links