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Ch. 4 - Exponential and Logarithmic Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 4 - Exponential and Logarithmic FunctionsProblem 17
Chapter 5, Problem 17

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log N-6

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1
Recognize that the expression is \( \log N^{-6} \), which involves a logarithm of a power.
Use the logarithmic property that \( \log a^b = b \log a \) to rewrite the expression.
Apply this property to get \( \log N^{-6} = -6 \log N \).
Since the expression is now expanded, check if \( \log N \) can be simplified further or evaluated without a calculator (usually it cannot unless \(N\) is a known value).
Conclude that the fully expanded form of the expression is \( -6 \log N \).

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Key Concepts

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

Properties of Logarithms

Properties of logarithms include rules such as the product, quotient, and power rules. These allow us to rewrite logarithmic expressions in simpler or expanded forms. For example, the power rule states that log(a^b) = b * log(a), which is essential for expanding expressions with exponents.
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Negative Exponents in Logarithms

A negative exponent indicates the reciprocal of the base raised to the positive exponent, i.e., a^{-b} = 1/a^b. When applied inside a logarithm, this affects the expression by introducing a negative sign outside the logarithm, as log(a^{-b}) = -b * log(a). Understanding this helps in correctly expanding and simplifying the expression.
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Evaluating Logarithms Without a Calculator

Evaluating logarithms without a calculator involves recognizing logarithms of numbers that are powers of the base or using known logarithmic values. Simplifying expressions using properties can reduce complex logs to simpler forms or constants, making it possible to evaluate them exactly without computational tools.
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Related Practice
Textbook Question

Graph each function by making a table of coordinates. If applicable, use a graphing utility to confirm your hand-drawn graph. f(x) = (0.6)x

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Textbook Question

In Exercises 19–24, the graph of an exponential function is given. Select the function for each graph from the following options:

f(x)=3x,g(x)=3x−1,h(x)=3x−1,f(x)=−3x,G(x)=3−x,H(x)=−3−x.f(x) = 3^x, \(\quad\) g(x) = 3^{x-1}, \(\quad\) h(x) = 3^x - 1, \(\f\)(x) = -3^x, \(\quad\) G(x) = 3^{-x}, \(\quad\) H(x) = -3^{-x}.

Textbook Question

Write each equation in its equivalent logarithmic form. b3 = 1000

Textbook Question

Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 4x=1/√2

Textbook Question

In Exercises 16–18, write each equation in its equivalent logarithmic form. 13^y = 874

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Textbook Question

Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 6(x−3)/4=√6