Ch. 4 - Exponential and Logarithmic Functions

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 4 - Exponential and Logarithmic Functions

Problem 130

Chapter 5, Problem 130

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

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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Related Practice

Textbook Question

In Exercises 125–128, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.

log_b (xy)⁵ = (log_b x + log_b y)⁵

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Textbook Question

Exercises 137–139 will help you prepare for the material covered in the next section. Solve for x: a(x - 2) = b(2x + 3)

Textbook Question

In Exercises 125–128, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.

$\frac{l o g_{7} 49}{l o g_{7} 7} = l o g_{7} 49 - l o g_{7} 7$

Textbook Question

Exercises 137–139 will help you prepare for the material covered in the next section. Solve: x(x - 7) = 3.

Textbook Question

Exercises 137–139 will help you prepare for the material covered in the next section. Solve: (x + 2)/(4x + 3) = 1/x

Textbook Question

In Exercises 125–128, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.

$l o g_{b} (x^{3} + y^{3}) = 3 l o g_{b} x + 3 l o g_{b} y$