Rules of Logarithms and Exponentials – Questions with Solutions

The rules of logarithmic and exponential functions are presented below. Their use in simplifying expressions and solving equations is illustrated through clear examples and fully worked solutions.

Throughout this page, we use the following notation:

Rules of Logarithmic Functions

Let \(U\) and \(V\) be positive real numbers, and let \(b > 0\), \(b \neq 1\).

  1. \( \log_b 1 = 0 \)
    Example: \( \log_5 1 = 0 \)
  2. \( \log_b(UV) = \log_b U + \log_b V \)
    Example: \( \log_2(4 \times 8) = \log_2 4 + \log_2 8 \)
  3. \( \log_b\!\left(\dfrac{U}{V}\right) = \log_b U - \log_b V \)
    Example: \( \log_2\!\left(\dfrac{8}{4}\right) = \log_2 8 - \log_2 4 \)
  4. \( \log_b(U^r) = r\log_b U \)
    Example: \( \log_3(9^2) = 2\log_3 9 \)
  5. \( \log_b(b^x) = x \)
    Example: \( \log_5(5^2) = 2 \)
  6. \( b^{\log_b x} = x, \; x > 0 \)
    Example: \( 2^{\log_2 5} = 5 \)
  7. \( \log_b U = \log_b V \iff U = V \)
    Example: \( \ln(x+1) = \ln(-x+4) \iff x+1 = -x+4 \)

Example 1

Simplify the following expressions using logarithmic rules:

  1. \( \log_9 1 \),
  2. \( \log_2(4 \times 16) \),
  3. \( \log_3\!\left(\dfrac{3}{27}\right) \),
  4. \( \log_4 4^6 \),
  5. \( \log_2 2^{x+1} \),
  6. \( e^{\ln(x^2+1)} \)

    Solution

    (a) \( \log_9 1 = 0 \)

    (b) \[ \log_2(4 \times 16) = \log_2 4 + \log_2 16 = \log_2 2^2 + \log_2 2^4 = 2 + 4 = 6 \]

    (c) \[ \log_3\!\left(\dfrac{3}{27}\right) = \log_3 3 - \log_3 27 = 1 - 3 = -2 \]

    (d) \( \log_4 4^6 = 6 \)

    (e) \( \log_2 2^{x+1} = x+1 \)

    (f) \( e^{\ln(x^2+1)} = x^2+1 \)

    Rules of Exponential Functions

    Let \(x\) and \(y\) be real numbers and let \(a > 0\).

    1. \( a^n = a \times a \times \cdots \times a \) (\(n\) times)
      Example: \( 2^4 = 16 \)
    2. \( a^0 = 1 \)
      Example: \( 100^0 = 1 \)
    3. \( a^{-x} = \dfrac{1}{a^x} \)
      Example: \( 6^{-2} = \dfrac{1}{6^2} \)
    4. \( a^x a^y = a^{x+y} \)
      Example: \( 4^2 \cdot 4^3 = 4^5 \)
    5. \( \dfrac{a^x}{a^y} = a^{x-y} \)
      Example: \( \dfrac{5^6}{5^4} = 5^2 \)
    6. \( (a^x)^y = a^{xy} \)
      Example: \( (7^2)^3 = 7^6 \)

    Example 2


    Use the rules of the exponential functions to simplify the following expressions
    1. \( 2^3 2^{-3} \)
    2. b) \( \dfrac{4^3 4^x}{4^{2x}} \)
    3. c) \( {(4^2)}^{2x} \)

    Solution

    1. \( 2^3 2^{-3} = 2^{3-3} = 3^0 = 1 \) , use rules 4 and 2.
    2. \( \dfrac{4^3 4^x}{4^{2x}} = \dfrac{4^{3+x}}{4^{2x}} = 4^{3+x-2x} = 4^{3-x} \) , use rules 4 and 5.
    3. \( {(4^2)}^{2x} = 4^{ 2 \times 2x } = 4^{4x} = (4^4)^{x} = 256^x \) , use rule 6

    Inverse Rule: Logarithmic and Exponential Functions

    Logarithmic and exponential functions are inverses of each other:

    \[ \log_b x = y \iff x = b^y \]

    Notes:

    Example 3

    Write each exponential equation in logarithmic form.

    1. \( 2^4 = 16 \),
    2. \( 3^3 = 27 \),
    3. \( e^5 = y \)

    Solution

    1. \( \log_2 16 = 4 \)
    2. \( \log_3 27 = 3 \)
    3. \( \ln y = 5 \) (since \(y = e^5\))

    Example 4

    Write each logarithmic equation in exponential form.

    1. \( \log_b 27 = x \),
    2. \( \log_{36} 6 = \tfrac12 \),
    3. \( \log_2\!\left(\tfrac18\right) = -3 \),
    4. \( \log_8 2 = \tfrac13 \)

    Solution

    1. \( 27 = b^x \)
    2. \( 6 = 36^{1/2} \)
    3. \( \tfrac18 = 2^{-3} \)
    4. \( 2 = 8^{1/3} \)

    Example 5

    Solve the equation \( \log_3 x = 4 \).

    Solution

    \[ x = 3^4 = 81 \]

    Example 6

    Solve each equation.

    1. \( \log_3 x = 5 \)
    2. \( \log_2(x-3) = 2 \)
    3. \( 2\log_3(-x+1) = 6 \)

    Solution

    1. \( x = 3^5 \)
    2. \( x-3 = 2^2 = 4 \Rightarrow x = 7 \)
    3. \( \log_3(-x+1) = 3 \Rightarrow -x+1 = 27 \Rightarrow x = -26 \)

    More References