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Logarithms (laws of logarithms, common logarithms, applications) — KCSE Mathematics

KCSE Mathematics · 105 practice questions · 4 syllabus objectives · 4 revision lessons

35 easy34 medium36 hard

Last updated · Aligned to the KNEC KCSE syllabus

What You'll Learn

Key learning outcomes for this topic, aligned to the KNEC KCSE syllabus.

Define a logarithm and convert between index form (aˣ = N) and logarithmic form (log_a N = x)

Apply the laws of logarithms (log AB = log A + log B, log A/B = log A – log B, log Aⁿ = n log A) to simplify and evaluate expressions

Use logarithm tables (or a calculator) to multiply, divide, find powers and roots of numbers; solve equations of the form aˣ = b

Logarithms (laws of logarithms, common logarithms, applications)

Revision Notes

Concise lesson notes for Logarithms (laws of logarithms, common logarithms, applications), written to the KCSE Mathematics marking standard. Read the first lesson free below.

Understanding Logarithms and Their Forms

A logarithm is the exponent to which a base must be raised to produce a given number. In mathematical terms, if we have an equation in index form:

aˣ = N,

it can be expressed in logarithmic form as:

log_a N = x.

Here, a is the base, N is the number, and x is the exponent.

Conversion Steps:

  1. Identify the base a.
  2. Recognize the number N.
  3. Determine the exponent x that satisfies the equation.

Example:

Convert the index form 2^3 = 8 into logarithmic form:

  • Here, a = 2, N = 8, and x = 3.
  • Thus, the logarithmic form is: log_2 8 = 3.

Understanding these conversions is crucial for solving logarithmic equations and applying logarithms in various mathematical contexts.

Key points to remember

  • A logarithm indicates the exponent of a base.
  • Convert aˣ = N to log_a N = x.
  • Identify base, number, and exponent for conversion.
  • Logarithmic form expresses the relationship clearly.
  • Common bases include 10 (common logarithm) and e (natural logarithm).

Worked example

Convert the index form 10^2 = 100 into logarithmic form: log_10 100 = 2.

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Lesson 2: Simplifying Expressions with Logarithms

Objective: Apply the laws of logarithms (log AB = log A + log B, log A/B = log A – log B, log Aⁿ = n log A) to simplify and evaluate expressions

Logarithms are powerful tools in mathematics, especially for simplifying expressions. The key laws of logarithms you should know are:

  • Product Law: ( \log(AB) = \log A + \log B )
  • Quotient Law: ( \log \left( \frac{A}{B} \right) = \log A - \log B )
  • Power Law: ( \log A^n = n \log A )

To apply these laws effectively, follow these steps:

  1. Identify parts of the expression that can be combined using the laws.
  2. Apply the relevant law to simplify the expression.
  3. If necessary, evaluate the logarithm using a calculator or known values.

For example, to simplify ( \log(10) + \log(5) ):

  • Using the Product Law:
    • ( \log(10) + \log(5) = \log(10 imes 5) = \log(50) )

Another example: simplify ( \log(100) - \log(10) ):

  • Using the Quotient Law:
    • ( \log(100) - \log(10) = \log \left( \frac{100}{10} \right) = \log(10) )

Remember, practice makes perfect!

  • Understand the three main laws of logarithms.
  • Use the Product Law to combine logarithms.
  • Apply the Quotient Law for subtraction of logarithms.
  • Utilize the Power Law for logarithmic expressions with exponents.

Simplify ( \log(8) + \log(4) - \log(2) ):

  • Using Product and Quotient Laws:
    • ( \log(8) + \log(4) = \log(32) )
    • ( \log(32) - \log(2) = \log(16) )
Lesson 3: Using Logarithms for Calculations

Objective: Use logarithm tables (or a calculator) to multiply, divide, find powers and roots of numbers; solve equations of the form aˣ = b

Logarithms are valuable tools for simplifying complex calculations. They allow us to multiply, divide, find powers, and roots easily. Here are the essential laws of logarithms:

  • Product Law: ( \log_b(MN) = \log_b(M) + \log_b(N) )
  • Quotient Law: ( \log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) )
  • Power Law: ( \log_b(M^p) = p \log_b(M) )

To use logarithm tables or a calculator for calculations, follow these steps:

  1. Multiplication: Use the Product Law to add logarithms.
  2. Division: Use the Quotient Law to subtract logarithms.
  3. Powers: Use the Power Law to multiply the logarithm by the exponent.
  4. Roots: Use the Power Law with fractional exponents.

For example, to multiply 100 and 1000:

  • Find ( \log_{10}(100) = 2 )
  • Find ( \log_{10}(1000) = 3 )
  • Add: ( \log_{10}(100 imes 1000) = 2 + 3 = 5 )
  • Therefore, ( 100 imes 1000 = 10^5 = 100000 ).
  • Logarithms simplify multiplication and division of large numbers.
  • Use logarithm tables for quick calculations.
  • Apply laws of logarithms for powers and roots.
  • Solve equations of the form aˣ = b using logarithms.

Solve for x: 10^x = 1000.

  • Take log base 10: x = log_{10}(1000) = 3.
Lesson 4: Understanding Logarithms and Their Laws

Objective: Logarithms (laws of logarithms, common logarithms, applications)

Logarithms are the inverse operations of exponentiation. The common logarithm (base 10) is often denoted as ( \log_{10}(x) ) or simply ( \log(x) ). Here are the laws of logarithms you should remember:

  • Product Law: ( \log_b(m imes n) = \log_b(m) + \log_b(n) )
  • Quotient Law: ( \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) )
  • Power Law: ( \log_b(m^n) = n \cdot \log_b(m) )

These laws allow you to simplify complex logarithmic expressions. For example, to evaluate ( \log(1000) ):

  1. Recognize that ( 1000 = 10^3 )
  2. Apply the Power Law: ( \log(1000) = \log(10^3) = 3 \cdot \log(10) = 3 \cdot 1 = 3 )

Logarithms are useful in solving exponential equations and in applications such as pH calculations in chemistry, sound intensity in decibels, and Richter scale for earthquakes.

  • Logarithms are the inverse of exponentiation.
  • Common logarithm is base 10, denoted as log(x).
  • Product, Quotient, and Power laws simplify expressions.
  • Logarithms are used in real-world applications.

Evaluate ( \log(100) + \log(10) ).

  • Apply Product Law: ( \log(100) + \log(10) = \log(100 \times 10) = \log(1000) = 3 ).

Sample Questions

Read 3 questions and answers free. Sign up to access all 105 questions with full KNEC-style marking schemes and a personalised study plan.

1
easySHORT ANSWER5 marks

In the context of mathematical analysis, the manipulation and application of logarithmic expressions play a crucial role in solving various equations. Consider the following problems that involve simplifying logarithmic terms, deriving values, and evaluating specific logarithmic expressions under defined constraints. (a) Simplify: log 18 + log 6 − log 3. (2 marks) (b) Given log 7 = 0.699, find log 7^4. (1 mark) (c) Evaluate: log₁₀(0.01) + log₁₀(10). (2 marks)

Answer & marking scheme

Part (a) — 2 marks
log(18×6/3) (1 mk)
Simplified value (1 mk)
Part (b) — 1 mark
log7^4=4×0.699 (1 mk)
Part (c) — 2 marks
log(0.01×10) (1 mk)
Correct value (1 mk)
2
easySHORT ANSWER3 marks

Explain how to use logarithm tables to find the product of 50 and 20. (3 marks)

Answer & marking scheme

Part (a) — 3 marks
Find log₁₀(50) from logarithm tables (1 mk)
Find log₁₀(20) from logarithm tables (1 mk)
Add the two logarithmic values and find the antilogarithm of the sum (1 mk)
3
easySHORT ANSWER2 marks

Name the value of x in the equation 10^x = 1000 using logarithms. (2 marks)

Answer & marking scheme

Part (a) — 2 marks
x = log₁₀(1000) (1 mk)
x = 3 (1 mk)
4

Name two properties of logarithms that can be used to simplify the expression log₁₀(50) + log₁₀(2). (2 marks)

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Frequently asked questions

What does the KCSE Mathematics topic "Logarithms (laws of logarithms, common logarithms, applications)" cover?

Logarithms (laws of logarithms, common logarithms, applications) covers Define a logarithm and convert between index form (aˣ = N) and logarithmic form (log_a N = x); Apply the laws of logarithms (log AB = log A + log B, log A/B = log A – log B, log Aⁿ = n log A) to simplify and evaluate expressions; Use logarithm tables (or a calculator) to multiply, divide, find powers and roots of numbers; solve equations of the form aˣ = b, and more, all aligned to the official KNEC KCSE Mathematics syllabus.

How many practice questions are available for Logarithms (laws of logarithms, common logarithms, applications)?

HighMarks has 105 Logarithms (laws of logarithms, common logarithms, applications) practice questions for KCSE Mathematics, each with a full marking scheme. The first 3 are free; sign up to access the rest, plus all KCSE mock exams and past papers.

Are these aligned with the KNEC KCSE syllabus?

Yes. Every objective on this page is taken directly from the official KNEC KCSE Mathematics syllabus. Practice questions match the KCSE exam format and are graded against the standard KNEC marking scheme.

How should I revise Logarithms (laws of logarithms, common logarithms, applications) for the KCSE exam?

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