3.7 F: Exponentials and Logarithms

• An exponential function increases rapidly if a > 1.


  True
• `log_a(1)` is equal to 0
• `log_a(m/n)` is equal to `log_a(m) - log_a(n)`
• What is the range of a logarithmic function?

  All real numbers
• `log_a(xy)` is equal to `log_a(x) + log_a(y)`
• What is the base of an exponential function?

  A positive constant
• The logarithm of a product is equal to the sum of the logarithms of its factors.


  True
• What are the four key properties of logarithms?

  Product, quotient, power, base
• Exponential functions either increase rapidly if `a > 1` or decrease rapidly if `0 < a < 1
• The formula for the change of base rule is log_b
• When converting log_2(8) to base 10, the formula yields log_10
• What is the quotient rule of logarithms?

  log_a(x/y) = log_a(x) - log_a(y)
• Steps for using logarithmic properties to simplify complex expressions

  1️⃣ Identify applicable logarithmic properties

  2️⃣ Apply the product, quotient, or power rule

  3️⃣ Combine or simplify terms

  4️⃣ Convert to exponential form if necessary

  5️⃣ Check for further simplification
• What is the general form of an exponential function?

  $f(x) =$$a^{x}$
• Logarithms are the inverses of exponential functions.

  True
• `log_a(mn)` is equal to `log_a(m) + log_a(n)`.


  True
• The domain of a logarithmic function is `x > 0
• The change of base formula for logarithms is `log_a(x) = log_b(x) / log_b(a)`.

  True
• The logarithm of a number `x` to the base `a` represents the exponent
• What is the change of base formula for logarithms?

  $log_{a}(x) =$$\frac{log_{b}(x)}{log_{b}(a)}$
• The change of base rule simplifies complex logarithmic calculations.


  True
• What is the general form of an exponential function?

  $f(x) =$$a^{x}$
• What is the primary purpose of the change of base rule for logarithms?

  Convert to a common base
• What is the result of converting log_2(8) to base 10 using the change of base rule?

  3
• The product rule states that the logarithm of a product is equal to the sum of the logarithms of the factors
• What is the change of base formula for logarithms?

  log_a(x) = log_b(x) / log_b(a)
• Exponential functions exhibit rapid growth or decay
• Steps for solving logarithmic equations

  1️⃣ Combine logarithms using properties

  2️⃣ Convert to exponential form

  3️⃣ Solve the algebraic equation

  4️⃣ Check solutions for validity
• log_a(b) = c is equivalent to a^c = b.


  True
• The rate of change of an exponential function is proportional to its value
• What is the value of `log_a(a)`?

  1
• What is the simplified form of `log_a(m^p)`?

  $p \log_{a}(m)$
• The logarithm of a number to the base `a` represents the exponent to which `a` must be raised to get that number.


  True
• What is the simplified form of `log_a(x/y)`?

  $log_{a}(x) - log_{a}(y)$
• What is the inverse of an exponential function?

  Logarithm
• Match the logarithmic property with its corresponding formula:

  Logarithm of a quotient ↔️ log_a(x/y) = log_a(x) - log_a(y)

  Logarithm of a power ↔️ log_a(x^n) = n*log_a(x)
• The change of base rule allows us to convert a logarithm from one base to another using base 10 or base e
• Logarithmic properties make complex calculations more straightforward.


  True
• The rate of change of an exponential function is proportional to the function's value.


  True
• The new base in the change of base rule is usually 10 or e.


  True