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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To get a grip on logarithms, let’s rewind to their historical context! The concept of logarithms was introduced by John Napier in the early 17th century as a way to simplify complex calculations. They were crucial for astronomers and navigators because they turned multiplication and division into simpler addition and subtraction, making computations easier before the era of calculators. For practical uses, think about how logarithms apply in fields like finance, where they help calculate compound interest, or in computer science for algorithms like binary search that work on logarithmic principles. Even in measuring sound intensity (decibels) or acidity (pH), logarithms show their versatility, allowing us to handle large ranges of values in a manageable way!