# 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?”

## Notes

Exponent Laws
• Review these laws from Algebra 1

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.

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

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

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

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

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

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!

• Change-of-Base Formula & Proof

• Examples

 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