Question

ch. 7 test form b
date 2/17/24 period 3
simplify. your answer should contain only positive exponents.

1. $\frac{a^{3}b^{6} cdot a^{5}b^{3}}{(a^{4}b^{-5})^{5}}$

rewrite each equation in exponential form.

1. $log_{6} \frac{1}{216} = -3$
2. $log_{9} 81 = 2$

rewrite each equation in logarithmic form.

1. $15^{2} = 225$
2. $324^{-\frac{1}{2}} = \frac{1}{18}$

evaluate each expression. use the change of base formula if needed. round to the 3rd decimal place.

1. $log_{7} \frac{1}{343}$
2. $log_{5} 23$

expand each logarithm.

1. $log_{9} left(c^{6}sqrt{a cdot b}

ight)$

1. $log \frac{zx^{2}}{3y^{5}}$

condense each expression to a single logarithm.

1. $4log_{2} a + 4log_{2} c - 20log_{2} b$
2. $log_{4} y + 2log_{4} z + log_{4} w + \frac{log_{4} x}{3}$

Explanation:

Step1: Simplify numerator exponents

Add exponents for like bases:
$a^{3+5}b^{6+3}=a^8b^9$

Step2: Simplify denominator exponents

Multiply exponents by power:
$(a^4b^{-5})^5=a^{4\times5}b^{-5\times5}=a^{20}b^{-25}$

Step3: Divide numerator by denominator

Subtract exponents for like bases:
$a^{8-20}b^{9-(-25)}=a^{-12}b^{34}$

Step4: Convert to positive exponents

Rewrite negative exponent as reciprocal:
$\frac{b^{34}}{a^{12}}$

Step5: Rewrite log to exponential form (2)

Use $\log_b x = y \iff b^y=x$:
$6^{-3}=\frac{1}{216}$

Step6: Rewrite log to exponential form (3)

Use $\log_b x = y \iff b^y=x$:
$9^2=81$

Step7: Rewrite exponential to log form (4)

Use $b^y=x \iff \log_b x = y$:
$\log_{15}225=2$

Step8: Rewrite exponential to log form (5)

Use $b^y=x \iff \log_b x = y$:
$\log_{324}\frac{1}{18}=-\frac{1}{2}$

Step9: Evaluate log expression (6)

Rewrite argument as power of base:
$\log_7 \frac{1}{343}=\log_7 7^{-3}=-3$

Step10: Evaluate log expression (7)

Use change of base formula $\log_b x=\frac{\ln x}{\ln b}$:
$\log_5 23=\frac{\ln 23}{\ln 5}\approx2.093$

Step11: Expand logarithm (8)

Use product/power rules, rewrite root as exponent:
$\log_9 c^6 + \log_9 (ab)^{\frac{1}{2}}=6\log_9 c + \frac{1}{2}\log_9 a + \frac{1}{2}\log_9 b$

Step12: Expand logarithm (9)

Use quotient/product/power rules:
$\log z + 2\log x - \log 3 - 5\log y$

Step13: Condense log expression (10)

Use power/quotient/product rules:
$4\log_2 ac - 20\log_2 b=\log_2 (ac)^4 - \log_2 b^{20}=\log_2 \frac{a^4c^4}{b^{20}}$

Step14: Condense log expression (11)

Use power/quotient/product rules:
$\log_4 y + \log_4 z^2 + \log_4 w + \log_4 x^{\frac{1}{3}}=\log_4 (ywz^2x^{\frac{1}{3}})=\log_4 (ywz^2\sqrt[3]{x})$

Answer:

1. $\frac{b^{34}}{a^{12}}$
2. $6^{-3}=\frac{1}{216}$
3. $9^2=81$
4. $\log_{15}225=2$
5. $\log_{324}\frac{1}{18}=-\frac{1}{2}$
6. $-3$
7. $2.093$
8. $6\log_9 c + \frac{1}{2}\log_9 a + \frac{1}{2}\log_9 b$
9. $\log z + 2\log x - \log 3 - 5\log y$
10. $\log_2 \frac{a^4c^4}{b^{20}}$
11. $\log_4 (ywz^2\sqrt[3]{x})$