Logarithms

What are Logarithms?

Logarithms are basically the "opposite" of exponents! They answer the question: "What power do I need to raise a base number to, to get another target number?"

Think about it like this:

  • Exponential Form: b^x = y
    • b is the base (the number being multiplied)
    • x is the exponent (the power)
    • y is the result
  • Logarithmic Form: \log_b(y) = x
    • \log stands for logarithm
    • b is still the base
    • y is the argument (the target number)
    • x is still the result (which is the exponent)

Example 1: Everyday Exponent to Logarithm

  • We know that 2^3 = 8
  • In logarithmic form, this means: \log_2(8) = 3
    • It reads: "log base 2 of 8 is 3."
    • It asks: "To what power do I raise 2 to get 8?" The answer is 3!

Example 2: Another Conversion

  • Given: 5^2 = 25
  • Logarithmic form: \log_5(25) = 2

Example 3: Logarithm to Exponent

  • Given: \log_{10}(100) = 2
  • Exponential form: 10^2 = 100

Types of Logarithms

  1. Common Logarithm (Base 10)

    • When you see \log(x) without a base written, it's always assumed to be base 10.
    • So, \log(x) = \log_{10}(x).
    • These are used a lot in science (like the pH scale or Richter scale).
    • Example 1: Find the value of \log(1000).
      • Step 1: Ask yourself, "10 to what power equals 1000?"
      • Step 2: We know 10 \times 10 \times 10 = 1000, so 10^3 = 1000.
      • Step 3: Therefore, \log(1000) = 3.

  2. Natural Logarithm (Base e)

    • This is a special logarithm where the base is a constant called Euler's number (e).
    • e is an irrational number, approximately 2.71828.
    • Instead of writing \log_e(x), we use the special notation \ln(x).
    • So, \ln(x) = \log_e(x).
    • Natural logarithms are very important in calculus, physics, and finance.
    • Example 1: Find the value of \ln(e^7).
      • Step 1: Ask yourself, "e to what power equals e^7?"
      • Step 2: It's clearly 7.
      • Step 3: Therefore, \ln(e^7) = 7.
    • Example 2: Convert \ln(5) = y to exponential form.
      • Step 1: Remember that \ln means base e.
      • Step 2: So, \log_e(5) = y.
      • Step 3: In exponential form, this is e^y = 5.

Key Properties of Logarithms

These rules help you simplify and solve equations involving logarithms.

  1. Product Rule

    • \logb(xy) = \logb(x) + \log_b(y)
    • This means the logarithm of a product is the sum of the logarithms.
    • Example: Expand \log_2(4 \cdot 8).
      • Step 1: Apply the product rule: \log2(4 \cdot 8) = \log2(4) + \log_2(8).
      • Step 2: Calculate each logarithm:
        • \log_2(4) = 2 (because 2^2 = 4)
        • \log_2(8) = 3 (because 2^3 = 8)
      • Step 3: Add the results: 2 + 3 = 5.
      • Check: \log2(4 \cdot 8) = \log2(32) = 5 (because 2^5 = 32). It works!

  2. Quotient Rule

    • \logb(\frac{x}{y}) = \logb(x) - \log_b(y)
    • The logarithm of a quotient is the difference of the logarithms.
    • Example: Condense \log3(27) - \log3(9).
      • Step 1: Apply the quotient rule: \log3(27) - \log3(9) = \log_3(\frac{27}{9}).
      • Step 2: Simplify the fraction: \log_3(3).
      • Step 3: Calculate the logarithm: \log_3(3) = 1 (because 3^1 = 3).

  3. Power Rule

    • \logb(x^p) = p \logb(x)
    • The logarithm of a number raised to a power is the power multiplied by the logarithm of the number.
    • Example: Simplify \log_5(25^3).
      • Step 1: Apply the power rule: \log5(25^3) = 3 \log5(25).
      • Step 2: Calculate the logarithm: \log_5(25) = 2 (because 5^2 = 25).
      • Step 3: Multiply: 3 \cdot 2 = 6.
      • Check: \log5(25^3) = \log5(15625) = 6 (because 5^6 = 15625). It works!

  4. Change of Base Formula

    • Sometimes you need to calculate a logarithm with a base your calculator doesn't have (most calculators only have \log for base 10 and \ln for base e).
    • \logb(x) = \frac{\logc(x)}{\log_c(b)}
    • You can choose any new base c (usually 10 or e).
    • Example: Calculate \log_3(7).
      • Step 1: Choose a new base, let's use base 10.
      • Step 2: Apply the formula: \log_3(7) = \frac{\log(7)}{\log(3)}.
      • Step 3: Using a calculator, \log(7) \approx 0.845 and \log(3) \approx 0.477.
      • Step 4: Divide: \frac{0.845}{0.477} \approx 1.77. So, \log_3(7) \approx 1.77.

Special Logarithm Values

  • \log_b(1) = 0
    • This is true because any non-zero base raised to the power of 0 equals 1 (b^0 = 1).
  • \log_b(b) = 1
    • This is true because any base raised to the power of 1 equals itself (b^1 = b).
  • b^{\log_b(x)} = x
    • This shows that exponential and logarithmic functions with the same base cancel each other out.
  • \log_b(b^x) = x
    • Another way to see that they cancel each other out.