By Fatskills Exam Guides Team — the exam nerds behind 28,500+ quizzes and 2.1M practice questions across 500+ global exams.
The change of base formula is a mathematical technique used to express a logarithm in terms of another base. It is a fundamental concept in mathematics and has numerous applications in various fields, including science, engineering, economics, and data analysis.
The change of base formula is crucial in situations where a specific base is not available or is not convenient to work with. For instance, in scientific notation, it is often more convenient to express numbers in terms of base 10, but sometimes it is necessary to work with other bases. The change of base formula allows us to convert between different bases, making it a powerful tool in mathematical calculations.
Express $\log_8(64)$ in terms of base 2.
$$ \log_8(64) = \frac{\log_2(64)}{\log_2(8)} $$ $$ = \frac{6}{3} $$ $$ = 2 $$
$\boxed{2}$
Express $\log_{10}(1000)$ in terms of base $e$.
$$ \log_{10}(1000) = \frac{\ln(1000)}{\ln(10)} $$ $$ = \frac{\ln(10^3)}{\ln(10)} $$ $$ = \frac{3\ln(10)}{\ln(10)} $$ $$ = 3 $$
$\boxed{3}$
Express $\log_5(125)$ in terms of base 10.
$$ \log_5(125) = \frac{\log_{10}(125)}{\log_{10}(5)} $$ $$ = \frac{\log_{10}(5^3)}{\log_{10}(5)} $$ $$ = \frac{3\log_{10}(5)}{\log_{10}(5)} $$ $$ = 3 $$
What is the change of base formula?
A) $\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$ B) $\log_b(a) = \log_c(a) + \log_c(b)$ C) $\log_b(a) = \log_c(a) - \log_c(b)$ D) $\log_b(a) = \log_c(a) \cdot \log_c(b)$
A) $\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$
The change of base formula is used to express a logarithm in terms of another base.
The distractors are tempting because they are similar to the correct answer, but with a small mistake.
What is the purpose of the change of base formula?
A) To simplify expressions B) To convert between different bases C) To find the inverse of a logarithm D) To find the square root of a number
B) To convert between different bases
The change of base formula is used to convert between different bases.
The distractors are tempting because they are related to the change of base formula, but not the main purpose.
What is the change of base formula for $\log_8(64)$ in terms of base 2?
A) $\frac{6}{3}$ B) $\frac{3}{6}$ C) $\frac{2}{3}$ D) $\frac{3}{2}$
A) $\frac{6}{3}$
The change of base formula is used to express the logarithm in terms of base 2.
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