Subtraction of Logarithms Practice Problems & Solutions

Master logarithm subtraction with same and different bases. Practice problems include step-by-step solutions using quotient rule and base change formula.

📚Master Logarithm Subtraction Through Targeted Practice
  • Apply the quotient rule: log_a(x) - log_a(y) = log_a(x/y) with identical bases
  • Convert logarithms with different bases using change of base formula
  • Solve complex logarithmic equations involving subtraction and simplification
  • Identify when to use quotient rule versus base conversion methods
  • Practice with real numbers like log_7(147) - log_7(3) step-by-step
  • Build confidence solving advanced problems with variables and mixed bases

Understanding Subtraction of Logarithms

Complete explanation with examples

Subtraction of Logarithms

The definition of a logarithm is:
logax=blog_a⁡x=b
X=abX=a^b

Where:
aa is the base of the exponent
XX is what appears inside the log, can also appear in parentheses
bb is the exponent we raise the log base to in order to obtain the number that appears inside of the log.


Subtraction of logarithms with identical base is based on the following rule:


logaxlogay=logaxylog_a⁡x-log_a⁡y=log_a⁡\frac{x}{y}

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.


Subtraction of logarithms with different bases is performed by changing the base using the following rule:

logaX=logbase we want to change toXlogbase we want to change toalog_aX=\frac{log_{base~we~want~to~change~to}X}{log_{base~we~want~to~change~to}a}

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.

Detailed explanation

Practice Subtraction of Logarithms

Test your knowledge with 6 quizzes

Calculate the value of the following expression:

\( \ln4\times(\log_7x^7-\log_7x^4-\log_7x^3+\log_2y^4-\log_2y^3-\log_2y) \)

Examples with solutions for Subtraction of Logarithms

Step-by-step solutions included
Exercise #1

log53log52= \log_53-\log_52=

Step-by-Step Solution

To solve the problem, we employ the property of logarithms for subtraction:

  • Step 1: Recognize the expression log53log52 \log_5 3 - \log_5 2 .
  • Step 2: Apply the logarithmic property for subtraction, logbalogbc=logb(ac) \log_b a - \log_b c = \log_b \left( \frac{a}{c} \right) .
  • Step 3: Substitute into the property: log53log52=log5(32) \log_5 3 - \log_5 2 = \log_5 \left( \frac{3}{2} \right) .

By applying the property, we simplify the expression to log532 \log_5 \frac{3}{2} . This is equivalent to log51.5 \log_5 1.5 . Therefore:

Therefore, the result of the expression is log51.5 \log_5 1.5 .

Answer:

log51.5 \log_51.5

Video Solution
Exercise #2

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 #3

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 #4

12log39log31.5= \frac{1}{2}\log_39-\log_31.5=

Step-by-Step Solution

To solve the problem 12log39log31.5 \frac{1}{2}\log_39-\log_31.5 , we need to apply the rules of logarithms:

  • **Step 1: Simplify with the power rule**
    Using the power rule 12log39=log391/2 \frac{1}{2}\log_39 = \log_39^{1/2} . Since 9=329 = 3^2, we have 91/2=321/2=319^{1/2} = 3^{2 \cdot 1/2} = 3^1. Thus, 12log39=log33=1\frac{1}{2}\log_39 = \log_3 3 = 1.
  • **Step 2: Apply the subtraction rule**
    Now, the expression becomes 1log31.51 - \log_3 1.5. Using the subtraction rule: 1log31.5=log33log31.5=log3(31.5)1 - \log_3 1.5 = \log_3 3 - \log_3 1.5 = \log_3 \left(\frac{3}{1.5}\right).
  • **Step 3: Simplify the fraction**
    Calculate 31.5\frac{3}{1.5}: it simplifies to 2 because 3÷1.5=23 \div 1.5 = 2.

Thus, the simplified expression is log32\log_3 2.

Using the provided answer choices, the correct answer matches choice log32 \log_3 2 , which corresponds to choice 2.

Therefore, the solution to the problem is log32 \log_3 2 .

Answer:

log32 \log_32

Video Solution
Exercise #5

14log61296log612log63= \frac{1}{4}\cdot\log_61296\cdot\log_6\frac{1}{2}-\log_63=

Step-by-Step Solution

We break it down into parts

log61296=x \log_61296=x

6x=1296 6^x=1296

x=4 x=4

144log612log63= \frac{1}{4}\cdot4\cdot\log_6\frac{1}{2}-\log_63=

log612log63= \log_6\frac{1}{2}-\log_63=

log6(12:3)=log616 \log_6\left(\frac{1}{2}:3\right)=\log_6\frac{1}{6}

log616=x \log_6\frac{1}{6}=x

6x=16 6^x=\frac{1}{6}

x=1 x=-1

Answer:

1 -1

Video Solution

Frequently Asked Questions

What is the rule for subtracting logarithms with the same base?

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When subtracting logarithms with identical bases, use the quotient rule: log_a(x) - log_a(y) = log_a(x/y). The first logarithm becomes the numerator and the second becomes the denominator inside a single logarithm.

How do you subtract logarithms with different bases?

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To subtract logarithms with different bases, first convert both to the same base using the change of base formula: log_a(x) = log_b(x)/log_b(a). Then apply the quotient rule for same-base subtraction.

What is the change of base formula for logarithms?

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The change of base formula is log_a(X) = log_c(X)/log_c(a), where 'c' is your desired new base. The numerator contains the original argument, and the denominator contains the original base.

Can you subtract log_3(x) - log_9(x) directly?

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No, you cannot subtract logarithms with different bases directly. First convert log_9(x) to base 3: log_9(x) = log_3(x)/log_3(9) = log_3(x)/2. Then subtract: log_3(x) - log_3(x)/2 = log_3(x)/2.

What does log_7(147) - log_7(3) equal step by step?

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Using the quotient rule: log_7(147) - log_7(3) = log_7(147/3) = log_7(49). Since 7² = 49, the answer is 2. This demonstrates how the subtraction rule can simplify complex-looking problems.

When should I use the quotient rule versus change of base?

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Use the quotient rule when logarithms have identical bases. Use change of base when bases are different - convert both to the same base (usually the smaller one), then apply the quotient rule.

What are common mistakes when subtracting logarithms?

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Common mistakes include: 1) Trying to subtract different bases directly, 2) Reversing numerator and denominator in the quotient rule, 3) Forgetting to simplify the fraction inside the logarithm, 4) Incorrectly applying the change of base formula.

How do I check my logarithm subtraction answers?

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Convert your final answer back to exponential form and verify. For example, if log_7(49) = 2, check that 7² = 49. You can also use a calculator to verify decimal approximations of your logarithmic expressions.

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