1 Express each of the following in the form $$ \log _{a} b=c $$. a $$ 10^{3}=1000=3 \quad $$ b $$ 3^{4}-81=4 $$ c $$ 256=2^{8} $$ d $$ 7^{0}=1 $$ c $$ 3^{-3}=\frac{1}{27}=-3 $$ f $$ 32^{-\frac{1}{1}}=\frac{1}{2}=\frac{1}{5} $$ g $$ 19^{1}=19 $$ h $$ 216=36^{1} $$ 2 Express each of the following using index notation. a $$ \log _{5} 125=3 $$ b $$ \log _{2} 16=4 $$ c $$ 5=\log _{10} 100000 $$ d $$ \log _{23} 1=0 $$ e $$ \frac{1}{2}=\log _{9} 3 $$ f $$ \lg 0.01=-2 $$ g $$ \log _{2} \frac{1}{8}=-3 $$ h $$ \log _{6} 6=1 $$ 3 Without using a calculator, find the exact value of a $$ \log _{7} 49 $$ b $$ \log _{4} 64 $$ c $$ \log _{2} 128 $$ d $$ \log _{3} 27 $$ e $$ \log _{5} 625 $$ f $$ \log _{8} 8 $$ g $$ \log _{7} 1 $$ h $$ \log _{15} \frac{1}{15} $$ i $$ \quad \log _{3} \frac{1}{9} $$ j $$ \quad \lg 0.001 $$ k $$ \log _{16} 2 $$ $$ 1 \log _{4} 8 $$ m $$ \log _{9} 243 $$ n $$ \log _{100} 0.001 $$ - $$ \log _{25} 125 $$ p $$ \log _{27} \frac{1}{9} $$ 4 Without using a calculator, find the exact value of $$ x $$ in each case. a $$ \log _{5} 25=x $$ b $$ \log _{2} x=6 $$ c $$ \log _{x} 64=3 $$ d $$ \lg x=-3 $$ e $$ \log _{x} 16=\frac{2}{3} $$ f $$ \log _{5} 1=x $$ g $$ \log _{x} 9=1 $$ h $$ \lg 10^{12}=x $$ i $$ 2 \log _{x} 7=1 $$ j $$ \log _{4} x=1.5 $$ k $$ \log _{x} 0.1=-\frac{1}{3} $$ l $$ 3 \log _{8} x+1=0 $$ 5 Express in the form $$ \log _{a} n $$ a $$ \log _{a} 4+\log _{a} 7 $$ b $$ \log _{a} 10-\log _{a} 5 $$. c $$ 2 \log _{a} 6 $$ d $$ \log _{a} 9-\log _{a} \frac{1}{3} $$ e $$ \frac{1}{2} \log _{a} 25+2 \log _{a} 3 $$ f $$ \log _{a} 48-3 \log _{a} 2-\frac{1}{2} \log _{a} 9 $$ 6 Express in the form $$ p \log _{q} x $$ a $$ \log _{q} x^{5} $$ b $$ \frac{1}{2} \log _{q} x^{15} $$ c $$ \log _{9} \frac{1}{x} $$ d $$ \log _{q} \sqrt[3]{x} $$ e $$ 4 \log _{q} \frac{1}{\sqrt{x}} $$ f $$ \log _{q} x^{2}+\log _{q} x^{5} $$ g $$ ^{\prime} \log _{q} \frac{1}{x^{2}}+\operatorname{loq}_{q} \frac{1}{x^{3}} $$ h $$ 3 \log _{9} x^{2}-\frac{1}{2} \log _{9} x^{4} $$ 7 Express in the form $$ \lg n $$ a $$ \lg 5+\lg 4 $$ b $$ \lg 12-\lg 6 $$ c $$ 3 \lg 2 $$ d $$ 4 \lg 3-\lg 9 $$ e $$ \frac{1}{2} \lg 16-\frac{1}{5} \lg 32 $$ f $$ 1+\lg 11 $$ g $$ \lg \frac{1}{50}+2 $$ h $$ 3-\lg 40 $$ 8 Without using a calculator, evaluate a $$ \log _{3} 54-\log _{3} 2 $$ b $$ \log _{5} 20+\log _{5} 1.25 $$ c $$ \log _{2} 16+\log _{3} 27 $$ d $$ \log _{6} 24+\log _{6} 9 $$ e $$ \log _{3} 12-\log _{3} 4 $$ f $$ \log _{4} 18-\log _{4} 9 $$ g $$ \log _{9} 4+\log _{9} 0.25 $$ h $$ 2 \lg 2+\lg 25 $$ i $$ \frac{1}{3} \log _{3} 8-\log _{3} 18 $$

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Let's break down the problem step by step, addressing each section in order.

### 1. Express each of the following in the form $$ \log _{a} b=c $$.

a. $$ 10^{3}=1000 $$ can be expressed as:
$$
\log_{10} 1000 = 3
$$

b. $$ 3^{4}=81 $$ can be expressed as:
$$
\log_{3} 81 = 4
$$

c. $$ 256=2^{8} $$ can be expressed as:
$$
\log_{2} 256 = 8
$$

d. $$ 7^{0}=1 $$ can be expressed as:
$$
\log_{7} 1 = 0
$$

e. $$ 3^{-3}=\frac{1}{27} $$ can be expressed as:
$$
\log_{3} \frac{1}{27} = -3
$$

f. $$ 32^{-\frac{1}{5}}=\frac{1}{2} $$ can be expressed as:
$$
\log_{32} \frac{1}{2} = -\frac{1}{5}
$$

g. $$ 19^{1}=19 $$ can be expressed as:
$$
\log_{19} 19 = 1
$$

h. $$ 216=36^{1} $$ can be expressed as:
$$
\log_{36} 216 = 1
$$

### 2. Express each of the following using index notation.

a. $$ \log_{5} 125=3 $$ can be expressed as:
$$
\log_{5} 5^{3} = 3
$$

b. $$ \log_{2} 16=4 $$ can be expressed as:
$$
\log_{2} 2^{4} = 4
$$

c. $$ 5=\log_{10} 100000 $$ can be expressed as:
$$
\log_{10} 10^{5} = 5
$$

d. $$ \log_{23} 1=0 $$ can be expressed as:
$$
\log_{23} 23^{0} = 0
$$

e. $$ \frac{1}{2}=\log_{9} 3 $$ can be expressed as:
$$
\log_{9} 3^{\frac{1}{2}} = \frac{1}{2}
$$

f. $$ \lg 0.01=-2 $$ can be expressed as:
$$
\log_{10} 10^{-2} = -2
$$

g. $$ \log_{2} \frac{1}{8}=-3 $$ can be expressed as:
$$
\log_{2} 2^{-3} = -3
$$

h. $$ \log_{6} 6=1 $$ can be expressed as:
$$
\log_{6} 6^{1} = 1
$$

### 3. Without using a calculator, find the exact value of

a. $$ \log_{7} 49 = 2 $$ because $$ 49 = 7^2 $$.

b. $$ \log_{4} 64 = 3 $$ because $$ 64 = 4^3 $$.

c. $$ \log_{2} 128 = 7 $$ because $$ 128 = 2^7 $$.

d. $$ \log_{3} 27 = 3 $$ because $$ 27 = 3^3 $$.

e. $$ \log_{5} 625 = 4 $$ because $$ 625 = 5^4 $$.

f. $$ \log_{8} 8 = 1 $$ because $$ 8 = 8^1 $$.

g. $$ \log_{7} 1 = 0 $$ because $$ 1 = 7^0 $$.

h. $$ \log_{15} \frac{1}{15} = -1 $$ because $$ \frac{1}{15} = 15^{-1} $$.

i. $$ \log_{3} \frac{1}{9} = -2 $$ because $$ \frac{1}{9} = 3^{-2} $$.

j. $$ \lg 0.001 = -3 $$ because $$ 0.001 = 10^{-3} $$.

k. $$ \log_{16} 2 = \frac{1}{4} $$ because $$ 2 = 16^{\frac{1}{4}} $$.

l. $$ \log_{4} 8 = \frac{3}{2} $$ because $$ 8 = 4^{\frac{3}{2}} $$.

m. $$ \log_{9} 243 = \frac{5}{2} $$ because $$ 243 = 9^{\frac{5}{2}} $$.

n. $$ \log_{100} 0.001 = -3 $$ because $$ 0.001 = 100^{-3} $$.

o. $$ \log_{25} 125 = \frac{3}{2} $$ because $$ 125 = 25^{\frac{3}{2}} $$.

p. $$ \log_{27} \frac{1}{9} = -\frac{2}{3} $$ because $$ \frac{1}{9} = 27^{-\frac{2}{3}} $$.

### 4. Without using a calculator, find the exact value of $$ x $$ in each case.

a. $$ \log_{5} 25=x $$ implies $$ x=2 $$ because $$ 25 = 5^2 $$.

b. $$ \log_{2} x=6 $$ implies $$ x=64 $$ because $$ 2^6 = 64 $$.

c. $$ \log_{x} 64=3 $$ implies $$ x=4 $$ because $$ 64 = 4^3 $$.

d. $$ \lg x=-3 $$ implies $$ x=0.001 $$ because $$ 10^{-3} = 0.001 $$.

e. $$ \log_{x} 16=\frac{2}{3} $$ implies $$ x=64 $$ because $$ 16 = 64^{\frac{2}{3}} $$.

f. $$ \log_{5} 1=x $$ implies $$ x=0 $$ because $$ 1 = 5^0 $$.

g. $$ \log_{x} 9=1 $$ implies $$ x=9 $$ because $$ 9 = 9^1 $$.

h. $$ \lg 10^{12}=x $$ implies $$ x=12 $$ because $$ 10^{12} $$ gives $$ 12 $$.

i. $$ 2 \log_{x} 7=1 $$ implies $$ x=49 $$ because $$ 7^{2} = 49
1. Express each of the following in the form $$ \log _{a} b=c $$: a. $$ \log_{10} 1000 = 3 $$ b. $$ \log_{3} 81 = 4 $$ c. $$ \log_{2} 256 = 8 $$ d. $$ \log_{7} 1 = 0 $$ e. $$ \log_{3} \frac{1}{27} = -3 $$ f. $$ \log_{32} \frac{1}{2} = -\frac{1}{5} $$ g. $$ \log_{19} 19 = 1 $$ h. $$ \log_{36} 216 = 1 $$ 2. Express each of the following using index notation: a. $$ \log_{5} 125 = 3 $$ → $$ 5^{3} = 125 $$ b. $$ \log_{2} 16 = 4 $$ → $$ 2^{4} = 16 $$ c. $$ 5 = \log_{10} 100000 $$ → $$ 10^{5} = 100000 $$ d. $$ \log_{23} 1 = 0 $$ → $$ 23^{0} = 1 $$ e. $$ \frac{1}{2} = \log_{9} 3 $$ → $$ 9^{\frac{1}{2}} = 3 $$ f. $$ \lg 0.01 = -2 $$ → $$ 10^{-2} = 0.01 $$ g. $$ \log_{2} \frac{1}{8} = -3 $$ → $$ 2^{-3} = \frac{1}{8} $$ h. $$ \log_{6} 6 = 1 $$ → $$ 6^{1} = 6 $$ 3. Without using a calculator, find the exact value of: a. $$ \log_{7} 49 = 2 $$ b. $$ \log_{4} 64 = 3 $$ c. $$ \log_{2} 128 = 7 $$

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