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9.7 – Properties of Exponents and Logarithms

Key Terms

  • Exponential Expression – An expression that involves an exponent.
    • The expression may contain variables, which may be the base or the exponent.
  • Logarithmic Expression – An expression that contains a logarithm.

Review

Remember…
  • Common logarithms are base 10.
  • Natural logarithms are base e (approximately 2.718).
  • A logarithm log_b a answers the question “How many times do you need to multiply b by itself to get a?”

Alg2B 9.7 Log Review

Notes

Exponent Laws
  • Review these laws from Algebra 1

Alg2B 9.7 Exp Product Law


Alg2B 9.7 Exp Quotient Law


Alg2B 9.7 Exp Power Law

 

Inverse Functions
  • Logarithms and exponents are inverses
    • For any positive number b not equal to 1 and any number or variable n, evaluate the following expression.
      • log_b(b^{n})=n.
    • For any positive number b not equal to 1 and any number or variable n, evaluate the following expression.
      • b^{log_b n}=n
    • The only logarithms you can find with a calculator are common and natural logarithms.

    • In any base, the logarithm of 1 always equals zero.

Alg2B 9.7 Inverse Logs

  • Example
    • log_3 (3^{log_3 x})=log_3 x

 

 Logarithm of a Product Property
  • Hint: multiplication goes with addition.

Alg2B 9.7 Product Property

 

Logarithm of a Quotient Property
  • Hint: division goes with subtraction.

Alg2B 9.7 Quotient Property

Alg2B 9.7 Log Quotient Property

 

Logarithm of a Power Property
  • log_b (a^{d})=d\bullet log_b a
  • Hint: move the exponent to the front!

Alg2B 9.7 Log Power Property

 

Logarithms that Equal 0 or 1
  • How many times do you need to multiply b by itself to get b?
    • Answer: 1
  • An exponent to the zero power equals 1; so, logs of ANY base to the number 1 equals 0.

Alg2B 9.7 Logs 0 or 1

 

Change-of-Base
  • The change-of-base formula changes the base of logarithms to 10 or e.
  • Both bases always give the same result.
  • Hint: base goes on the bottom!

Alg2B 9.7 Change of Base2


  • Change-of-Base Formula & Proof

Alg2B 9.7 Change of Base Proof


  • Examples

Alg2B 9.7 Change of Base Ex

Alg2B 9.7 Change of Base

  • Logarithm Formulas
    • ln(e^{x})=x
    • e^{(ln\; x)}=x
  • Logarithm of a Product
    • log_b (a\bullet d)=log_b a+log_b d
  • Logarithm of a Quotient
    • log_b (\frac{a}{d})=log_b a-log_b d
  • Logarithm of a Power
    • log_b (a^{d})=d\bullet log_b a
  • Logarithm Equal to 0
    • log_b 1=0
  • Logarithm Equal to 1
    • log_b b=1

Examples

  • Ex 1. Which expressions are equivalent to the one below?
  • 3^{4}\bullet 3^{x}
    • Answers: 81\bullet 3^{x} and 3^{4+x}
  • Ex 2. Which expressions are equivalent to the one below?
  • 16x
    • Answers: 4^{2x}, 4^{x}\bullet 4^{x}, and (4\bullet 4)^{x}
  • Ex 3. Which expressions are equivalent to the one below?
  • \frac{21^{x}}{7^{x}}
    • Answers: 3^{x}, \frac{7^{x}\bullet 3^{x}}{7^{x}}, and (\frac{21}{7})^{x}
  • Ex 4. Which expressions are equivalent to the one below?
  • 3x
    • Answers: (\frac{18}{6})^{x}, \frac{18^{x}}{6^{x}}, and 3\bullet 3^{x-1}
  • Ex 5. Which expression is equivalent to b^{m}\bullet b^{n}?
    • Answer: b^{m+n}
  • Ex 6. Which expression is equivalent to (b^{n})^{m}?
    • Answer: b^{m\bullet n}
  • Ex 7. Which expressions are equivalent to the one below?
  • log 2 – log 8
    • Answers: log(2)+log(\frac{1}{8}) and log(\frac{1}{4})
  • Ex 8. Which expressions are equivalent to the one below?
  • log_5 5+log_5 125
    • Answers: 4, log_5 (5^{4}), and log_5 625
  • Ex 9. Which expressions are equivalent to the one below?
  • ln(e^{5})
    • Answers: 5 and 5\bullet ln\; e
  • Ex 10. Which expressions are equivalent to the one below?
  • log(10^{3})
    • Answers: 3\bullet log 10 and 3
  • Ex 11. Which expressions are equivalent to the one below?
  • log_9 1\bullet log_9 81
    • Answer: 0
  • Ex 12. Simplify the following expression.
  • log(x^{4})-log(x^{3})
    • Answer: log(x)
  • Ex 13. Simplify the following expression.
  • log(9x^{5})+5\; log(\frac{1}{x})
    • Answer: Log 9
  • Ex 14. Evaluate the following expression. Round your answer to two decimal places.
  • log_6 8
    • Answer: 1.16
  • Ex 15. Evaluate the following expression. Round your answer to two decimal places.
  • log_6 e
    • Answer: 0.56
  • Ex 16. Evaluate the following expression. You should do this problem without a calculator.
  • e^{ln\; 5}
    • Answer: 5
  • Ex 17. Evaluate the following expression. You should do this problem without a calculator.
  • ln\; e^{e}
    • Answer: e
  • Ex 18. Evaluate the following expression.
  • log_8 64
    • Answer: 2
  • Ex 19. Evaluate the following expression. You should do this problem without a calculator.
  • log_6 6
    • Answer: 1

 

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