Question

1. $\log_{32}16 = \square$ 11. simplify the expression. $\log_{7}7^{-3x} = \square$ 12. rewrite $\log_{3}9 = 2$ in exponential form. $\square = \square$ 13. rewrite the equation $6^{2} = 36$ in logarithmic form. $\square = \square$

Explanation:

Step1: Solve $\log_{32}16$

Express 32 and 16 as powers of 2: $32=2^5$, $16=2^4$. Use log rule $\log_{a^m}a^n=\frac{n}{m}$.
$\log_{2^5}2^4=\frac{4}{5}=0.8$

Step2: Simplify $\log_{7}7^{-3x}$

Use log identity $\log_{a}a^k=k$.
$\log_{7}7^{-3x}=-3x$

Step3: Rewrite $\log_{3}9=2$ to exponential form

Use log definition: $\log_{a}b=c \iff a^c=b$.
$3^2=9$

Step4: Rewrite $6^2=36$ to logarithmic form

Use exponent to log definition: $a^c=b \iff \log_{a}b=c$.
$\log_{6}36=2$

Answer:

1. $\log_{32}16 = 0.8$
2. $\log_{7}7^{-3x} = -3x$
3. $3^2=9$
4. $\log_{6}36=2$