Rules of Logarithms and Exponentials

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.

Notation Guide:

Table of Contents

Rules of Logarithmic Functions

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

  1. Logarithm of 1: \( \log_b 1 = 0 \)
    Example: \( \log_5 1 = 0 \)
  2. Logarithm of the Base: \( \log_b b = 1 \)
    Example: \( \log_7 7 = 1 \)
  3. Product Rule: \( \log_b(UV) = \log_b U + \log_b V \)
    Example: \( \log_2(4 \times 8) = \log_2 4 + \log_2 8 \)
  4. Quotient Rule: \( \log_b\left(\dfrac{U}{V}\right) = \log_b U - \log_b V \)
    Example: \( \log_2\left(\dfrac{8}{4}\right) = \log_2 8 - \log_2 4 \)
  5. Power Rule: \( \log_b(U^r) = r\log_b U \)
    Example: \( \log_3(9^2) = 2\log_3 9 \)
  6. Identity Rule: \( \log_b(b^x) = x \)
    Example: \( \log_5(5^2) = 2 \)
  7. Exponentiation of Log: \( b^{\log_b x} = x, \; x \gt 0 \)
    Example: \( 2^{\log_2 5} = 5 \)
  8. Equality Rule: \( \log_b U = \log_b V \iff U = V \)
    Example: \( \ln(x+1) = \ln(-x+4) \iff x+1 = -x+4 \)
  9. Reciprocal Rule: \( \log_b\left(\dfrac{1}{U}\right) = -\log_b U \)
    Example: \( \log_3(1/9) = -\log_3 9 = -2 \)

Example 1: Simplify Logarithmic Expressions

Show Solution

Question: 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

  1. Using Rule 1: \( \log_9 1 = 0 \)
  2. Using the Product Rule and Power Rule: \[ \log_2(4 \times 16) = \log_2 4 + \log_2 16 = \log_2(2^2) + \log_2(2^4) = 2 + 4 = 6 \]
  3. Using the Quotient Rule: \[ \log_3\left(\dfrac{3}{27}\right) = \log_3 3 - \log_3 27 = 1 - 3 = -2 \]
  4. Using the Identity Rule: \( \log_4 4^6 = 6 \)
  5. Using the Identity Rule: \( \log_2 2^{x+1} = x+1 \)
  6. Using the Exponentiation Rule: \( e^{\ln(x^2+1)} = x^2+1 \)

Verification: \( \log_2 64 = 6 \) ✓ (since \( 2^6 = 64 \)) and \( \log_3(1/9) = -2 \) ✓ (since \( 3^{-2} = 1/9 \))

Rules of Exponential Functions

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

  1. Definition: \( a^n = a \times a \times \cdots \times a \) (\( n \) times)
    Example: \( 2^4 = 16 \)
  2. Zero Exponent: \( a^0 = 1 \)
    Example: \( 100^0 = 1 \)
  3. Negative Exponent: \( a^{-x} = \dfrac{1}{a^x} \)
    Example: \( 6^{-2} = \dfrac{1}{6^2} = \dfrac{1}{36} \)
  4. Product Rule: \( a^x a^y = a^{x+y} \)
    Example: \( 4^2 \cdot 4^3 = 4^5 \)
  5. Quotient Rule: \( \dfrac{a^x}{a^y} = a^{x-y} \)
    Example: \( \dfrac{5^6}{5^4} = 5^2 \)
  6. Power of a Power: \( (a^x)^y = a^{xy} \)
    Example: \( (7^2)^3 = 7^6 \)
  7. Power of a Product: \( (ab)^x = a^x b^x \)
    Example: \( (2 \cdot 3)^4 = 2^4 \cdot 3^4 \)

Example 2: Simplify Exponential Expressions

Show Solution

Question: Use the rules of exponential functions to simplify the following expressions:

  1. \( 2^3 \cdot 2^{-3} \)
  2. \( \dfrac{4^3 \cdot 4^x}{4^{2x}} \)
  3. \( (4^2)^{2x} \)

Solution

  1. Use the Product Rule and Zero Exponent: \[ 2^3 \cdot 2^{-3} = 2^{3-3} = 2^0 = 1 \]
  2. Use the Product and Quotient Rules: \[ \dfrac{4^3 \cdot 4^x}{4^{2x}} = \dfrac{4^{3+x}}{4^{2x}} = 4^{3+x-2x} = 4^{3-x} \]
  3. Use the Power of a Power Rule: \[ (4^2)^{2x} = 4^{2 \times 2x} = 4^{4x} = (4^4)^x = 256^x \]

Verification: \( 4^{3-x} \cdot 4^{2x} = 4^{3+x} = 4^3 \cdot 4^x \) ✓

Inverse Rule: Logarithmic and Exponential Functions

Logarithmic and exponential functions are inverses of each other.

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

Notes:

  • The bases of the logarithmic and exponential forms are identical.
  • The logarithm gives the exponent to which the base must be raised to obtain the specified number.
  • This relationship is the bridge between solving logarithmic and exponential equations.

Example 3: Convert Exponentials to Logarithms

Show Solution

Question: Write each 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: Convert Logarithms to Exponentials

Show Solution

Question: Write each equation in exponential form.

  1. \( \log_b 27 = x \)
  2. \( \log_{36} 6 = \frac{1}{2} \)
  3. \( \log_2\left(\frac{1}{8}\right) = -3 \)
  4. \( \log_8 2 = \frac{1}{3} \)

Solution

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

Example 5: Solve a Single Logarithmic Equation

Show Solution

Question: Solve \( \log_3 x = 4 \).

Solution

Rewrite in exponential form using the inverse rule:

\[ x = 3^4 = 81 \]

Verification: \( \log_3(81) = \log_3(3^4) = 4 \) ✓

Example 6: Solve Multiple Logarithmic Equations

Show Solution

Question: Solve each equation.

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

Solution

  1. Rewrite in exponential form: \[ x = 3^5 = 243 \]
    Verification: \( \log_3(243) = 5 \) ✓
  2. Rewrite in exponential form and solve: \[ x - 3 = 2^2 \] \[ x - 3 = 4 \implies x = 7 \]
    Verification: \( \log_2(7-3) = \log_2(4) = 2 \) ✓ and \( x - 3 = 4 \gt 0 \) ✓
  3. Divide by 2, rewrite, and solve: \[ \log_3(-x+1) = 3 \] \[ -x + 1 = 3^3 = 27 \implies x = -26 \]
    Verification: \( \log_3(-(-26)+1) = \log_3(27) = 3 \) ✓ and \( -x + 1 = 27 \gt 0 \) ✓

Example 7: Solve an Exponential Equation Using Logs

Show Solution

Question: Solve \( 5^{2x-1} = 125 \) for \( x \).

Solution

Method 1 — Equate exponents directly: Since \( 125 = 5^3 \):

\[ 2x - 1 = 3 \implies 2x = 4 \implies x = 2 \]

Method 2 — Convert using logarithms:

\[ \log_5(125) = 2x - 1 \implies 3 = 2x - 1 \implies x = 2 \]

Verification: \( 5^{2(2)-1} = 5^3 = 125 \) ✓

Quick Reference Summary

Operation Logarithm Rule Exponential Rule
Product\( \log_b(UV) = \log_b U + \log_b V \)\( a^x a^y = a^{x+y} \)
Quotient\( \log_b(U/V) = \log_b U - \log_b V \)\( a^x / a^y = a^{x-y} \)
Power\( \log_b(U^r) = r \log_b U \)\( (a^x)^y = a^{xy} \)
Zero\( \log_b 1 = 0 \)\( a^0 = 1 \)
Reciprocal\( \log_b(1/U) = -\log_b U \)\( a^{-x} = 1/a^x \)
Inverse\( \log_b(b^x) = x \) and \( b^{\log_b x} = x \)\( \log_b x = y \iff x = b^y \)

Common Mistakes to Avoid

  1. Assuming \( \log_b(U + V) = \log_b U + \log_b V \). The product rule applies to a product, not a sum.
  2. Applying the power rule to the base. \( \log_b(U^r) = r \log_b U \) applies to the argument \( U \), not to \( b \).
  3. Forgetting the domain. After solving a logarithmic equation, always verify that the argument is strictly positive.
  4. Dividing by \( \log_b \) or \( \ln \) as if it were a number. \( \log_b \) is a function, not a variable to be cancelled.
  5. Using \( b = 1 \) as a base. \( \log_1(x) \) is undefined.
  6. Confusing \( \log \) with \( \ln \). \( \log \) usually means base 10; \( \ln \) means base \( e \).

Practice Exercises

Try each exercise on your own before viewing the answer.

Simplify Using Logarithm Rules
  1. \( \log_5(25) + \log_5(5) \)
  2. \( \log_2(32) - \log_2(4) \)
  3. \( \log_3(81^2) \)
  4. \( 10^{\log_{10}(7)} \)
Simplify Using Exponential Rules
  1. \( 3^4 \cdot 3^{-2} \)
  2. \( \dfrac{7^5}{7^3} \)
  3. \( (2^3)^4 \)
  4. \( 5^{0} \cdot 5^{-1} \)
View All Answers

Logarithm exercises:

  1. \( \log_5 25 + \log_5 5 = 2 + 1 = 3 \)
  2. \( \log_2 32 - \log_2 4 = 5 - 2 = 3 \)
  3. \( \log_3(81^2) = 2 \log_3 81 = 2 \times 4 = 8 \)
  4. \( 10^{\log_{10} 7} = 7 \)

Exponential exercises:

  1. \( 3^4 \cdot 3^{-2} = 3^{4-2} = 3^2 = 9 \)
  2. \( \dfrac{7^5}{7^3} = 7^{5-3} = 7^2 = 49 \)
  3. \( (2^3)^4 = 2^{12} = 4096 \)
  4. \( 5^0 \cdot 5^{-1} = 1 \cdot \dfrac{1}{5} = \dfrac{1}{5} \)

Frequently Asked Questions

What are the main rules of logarithms?

The main rules of logarithms are: (1) Product rule: \( \log_b(UV) = \log_b U + \log_b V \). (2) Quotient rule: \( \log_b(U/V) = \log_b U - \log_b V \). (3) Power rule: \( \log_b(U^r) = r\log_b U \). (4) \( \log_b 1 = 0 \). (5) \( \log_b b = 1 \). (6) \( \log_b(b^x) = x \). (7) \( b^{\log_b x} = x \).

What are the main rules of exponentials?

The main rules of exponentials are: (1) Product rule: \( a^x a^y = a^{x+y} \). (2) Quotient rule: \( a^x / a^y = a^{x-y} \). (3) Power of a power: \( (a^x)^y = a^{xy} \). (4) Zero exponent: \( a^0 = 1 \). (5) Negative exponent: \( a^{-x} = 1/a^x \).

How do you convert between logarithmic and exponential form?

The inverse relationship is: \( \log_b(x) = y \) if and only if \( x = b^y \). The base \( b \) stays the same; the exponent \( y \) and the argument \( x \) swap roles between the two forms.

What is the difference between the product rule and the power rule for logarithms?

The product rule applies to a product inside a logarithm: \( \log_b(UV) = \log_b U + \log_b V \). The power rule applies to an exponent on the argument: \( \log_b(U^r) = r\log_b U \). They are often used together to expand or condense logarithmic expressions.

Can you apply the product rule to \( \log_b(U + V) \)?

No. The product rule only applies to a product of arguments, not a sum. \( \log_b(U + V) \) cannot be simplified using the standard logarithm rules.

How do you solve an equation like \( \log_3(x) = 4 \)?

Convert the logarithmic equation to exponential form using the inverse rule: \( \log_b(x) = y \iff x = b^y \). For \( \log_3(x) = 4 \), rewrite as \( x = 3^4 = 81 \). Verify that \( x \) satisfies the domain (\( x \gt 0 \)).

Why must the base of a logarithm be positive and not equal to 1?

The base \( b \) must satisfy \( b \gt 0 \) and \( b \neq 1 \). If \( b \) were negative, expressions like \( b^x \) would be undefined for non-integer exponents. If \( b = 1 \), then \( b^x = 1 \) for all \( x \), so \( \log_1 \) would not be a well-defined function.

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