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

   • $\log<em>b(xy) = \log</em>b(x) + \log_b(y)$
   • 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: $\log<em>2(4 \cdot 8) = \log</em>2(4) + \log_2(8)$.
     • 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: $\log<em>2(4 \cdot 8) = \log</em>2(32) = 5$ (because $2^5 = 32$). It works!

2. Quotient Rule

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

3. Power Rule

   • $\log<em>b(x^p) = p \log</em>b(x)$
   • 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: $\log<em>5(25^3) = 3 \log</em>5(25)$.
     • Step 2: Calculate the logarithm: $\log_5(25) = 2$ (because $5^2 = 25$).
     • Step 3: Multiply: $3 \cdot 2 = 6$.
     • Check: $\log<em>5(25^3) = \log</em>5(15625) = 6$ (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 $e$).
   • $\log<em>b(x) = \frac{\log</em>c(x)}{\log_c(b)}$
   • 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.