General Mathematics WAEC

Logarithms

Visão Geral

Logarithms are an essential concept in mathematics that allow us to simplify complex calculations involving exponents, making computations more manageable and efficient. Understanding the relationship between logarithms and indices is fundamental in solving a wide range of mathematical problems.

Relationship Between Indices and Logarithms: One of the key objectives in studying logarithms is to establish a clear understanding of how they relate to indices. When we have an exponential equation in the form of \(y = a^x\), we can rewrite it in logarithmic form as \(\log_a y = x\). This relationship, often denoted as \(y = a^x \implies \log_a y = x\), forms the basis for converting between exponential and logarithmic expressions.

By converting between these forms, we can simplify calculations involving very large or very small numbers, as logarithms condense these numbers into more manageable values. The concept of logarithms is particularly useful in scientific calculations, where dealing with numbers in standard form (scientific notation) is common practice.

Basic Rules of Logarithms: In addition to understanding the relationship between logarithms and indices, it is crucial to grasp the basic rules that govern logarithmic operations. These rules include:

  1. Addition Rule: \(\log_a (P \cdot Q) = \log_a P + \log_a Q\)
  2. Subtraction Rule: \(\log_a (P / Q) = \log_a P - \log_a Q\)
  3. Exponent Rule: \(\log_a P^N = N \cdot \log_a P\)

These rules are essential for simplifying logarithmic expressions and solving equations involving logarithms efficiently. By applying these rules, we can break down complex logarithmic terms into simpler components, facilitating accurate calculations in various mathematical contexts.

Moreover, understanding the basic rules of logarithms enables us to manipulate logarithmic expressions effectively, allowing us to solve a wide range of problems across different areas of mathematics and scientific disciplines.

Objetivos

  1. Understand the relationship between indices and logarithms
  2. Utilize logarithmic tables and antilogarithms effectively
  3. Apply basic rules of logarithms in mathematical calculations

Nota de Aula

Não Disponível

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  1. Expand the following logarithmic expression: log10(2^3) - log10√100 A. 1 B. 2 C. 3 D. 4 Answer: B. 2
  2. Simplify the following expression: log5(125) + log5(25) - log5(5) A. 1 B. 2 C. 3 D. 4 Answer: A. 1
  3. If log2(x) = 3, what is the value of x? A. 4 B. 6 C. 8 D. 16 Answer: D. 16
  4. Evaluate log5(625) - log5(5) A. 2 B. 3 C. 4 D. 5 Answer: A. 2
  5. What is the value of log3(27) + log3(9) - log3(3)? A. 2 B. 3 C. 4 D. 5 Answer: D. 5
  6. If log10(x) = 2.5, what is the value of x in standard form (scientific notation)? A. 3.16 x 10^2 B. 3.16 x 10^3 C. 3.16 x 10^4 D. 3.16 x 10^5 Answer: C. 3.16 x 10^4
  7. What is the result of log5(125) - log5(5)? A. 1 B. 2 C. 3 D. 4 Answer: B. 2
  8. Given loga(b) = c, what is b in terms of a and c? A. a^c B. a/c C. a + c D. a - c Answer: A. a^c
  9. Simplify: log3(81) - log3(9) A. 1 B. 2 C. 3 D. 4 Answer: B. 2
  10. If log2(x) = 5 and log2(y) = 3, what is log2(x/y)? A. 2 B. 3 C. 4 D. 5 Answer: D. 5

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Questões de revisão

Pergunta-se como são as perguntas anteriores sobre este tópico? Aqui estão várias perguntas sobre Logarithms de anos passados.

JAMB UTME - Mathematics - 2025

Pergunta 1 Relatório

Simplify - log\(_{10}\) 0.00001.

Detalhes da Resposta

Logarithms help us answer the question: "To what power must we raise a certain base to get a particular number?" In this case, we are working with base 10 logarithms (log\(_{10}\)). The expression given is \(- \log_{10} 0.00001\).


First, recall the definition:

  • \(\log_{10} x\) is the power to which 10 must be raised to get \(x\).

Let's break it down:

  • What is \(0.00001\) in terms of powers of 10? We can write \(0.00001\) as \(10^{-5}\), because:
\[ 10^{-5} = \frac{1}{10^5} = 0.00001 \]

Now use the property of logarithms that says \(\log_{10}(10^a) = a\):

\[ \log_{10}(0.00001) = \log_{10}(10^{-5}) = -5 \]

But the original question asks for the negative of this value:

\[ - \log_{10}(0.00001) = -(-5) = 5 \]

Therefore, the simplified value is \(5\).


  • This is because the logarithm gives the exponent: \(10^{-5} = 0.00001\), so the log is \(-5\). The negative sign outside gives us \(5\).
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NECO SSCE - General Mathematics - 2008

Pergunta 1 Relatório

Find the value of log\(_{\sqrt{3}}\) 81
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WAEC SSCE - General Mathematics - 2022

Pergunta 1 Relatório

Given that \( \log_{3} 27 = 2x + 1 \), find the value of x.

Detalhes da Resposta

Recall that: log3 3  27 → log3 3 33

3log3 3 3 → 3 * 1

= 3

Then log3 3  27 = 2x + 1

→ 3 = 2x + 1 

3 - 1 = 2x

2 = 2x

1 = x

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