Ch. 4 - Exponential and Logarithmic Functions

Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경

Blitzer 8th Edition

Ch. 4 - Exponential and Logarithmic Functions

문제 130

5장, 문제 130

If log 3 = A and log 7 = B, find log7 (9) in terms of A and B.

검증된 단계별 안내

Recall the change of base formula for logarithms:

\log a_{b} = \frac{\log}{c} \frac{\log}{c}

, which means

\log a_{b} = \frac{\log}{c} \frac{\log}{c}

for any positive base

c

not equal to 1.

Apply the change of base formula to

\log 7_{9}

, choosing base 10 (common logarithm) for convenience:

\log 7_{9} = \frac{\log}{10} \frac{\log}{10}

Express

\log 9

in terms of

\log 3

using the property

\log a^n = n \log a

. Since

9 = 3^2

, we have

\log 9 = 2 \log 3

Substitute the given values:

\log 3 = A

and

\log 7 = B

into the expression from step 2 and 3, so that

\log 7_{9} = 2A B

Write the final expression for

\log 7_{9}

in terms of A and B as

\log 7_{9} = 2A B

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Change of Base Formula

The change of base formula allows you to rewrite logarithms with any base in terms of logarithms with a different base, typically base 10 or e. It states that log_b(x) = log_c(x) / log_c(b), which is essential for expressing log base 7 of 9 in terms of log base 10 values A and B.

Properties of Logarithms

Logarithmic properties such as log(xy) = log x + log y and log(x^n) = n log x help simplify expressions. Recognizing that 9 = 3^2 allows rewriting log_7(9) as log_7(3^2) = 2 log_7(3), facilitating the use of given values A and B.

Expressing Logarithms in Terms of Given Variables

Given log 3 = A and log 7 = B, these represent logarithms with a common base (usually 10). Using the change of base formula, log_7(3) can be expressed as log(3)/log(7) = A/B, enabling the expression of log_7(9) in terms of A and B.