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Chapter 4: Problem 122

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because logarithms are exponents, the product, quotient, and power rules remind me of properties for operations with exponents.

Short Answer

Expert verified
The statement makes sense. Logarithms are a reverse way of expressing exponents and share similar properties such as the product, quotient, and power rules.

Step by step solution

01

Understanding the relationship between logarithms and exponents

Logarithms and exponents are closely related. If \(a^b = c\), then the base \(a\) logarithm of \(c\) is \(b\), which can be represented as \( log_{a}c = b \). This relationship shows that logarithms are essentially exponents.
02

Explain the Product Rule for Logarithms and its analogy to Exponents

The product rule of logarithms states that the logarithm of a product is equal to the sum of the logarithms of its factors. \( log_b(xy) = log_bx + log_by \). This is analogous to the rule for exponents where the product of bases to exponents is equal to the sum of the exponents (i.e., \( a^m * a^n = a^{(m+n)}\)).
03

Explain the Quotient Rule for Logarithms and its analogy to Exponents

The quotient rule of logarithms states that the logarithm of a quotient is equal to the logarithm of the numerator minus the logarithm of the denominator: \( log_b(x/y) = log_bx - log_by \). This is analogous to the rule for exponents where the quotient of the bases to exponents is equal to the subtraction of the exponents (i.e., \( a^m / a^n = a^{(m-n)}\)).
04

Explain Power Rule for Logarithms and its analogy to Exponents

The power rule of logarithms states that the logarithm of a base to exponent is equal to the product of the exponent and the logarithm of the base: \( log_b(x^n) = n * log_bx \). This is analogous to the rule for exponents where the power to an exponent can be multiplied (i.e., \( (a^m)^n = a^{m*n} \)).

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Most popular questions from this chapter

In Exercises 139鈥�142, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \(\log _{b} x\) is the exponent to which \(b\) must be raised to obtain \(x\).

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$ 2^{x+1}=8 $$

Explain the differences between solving \(\log _{3}(x-1)=4\) and \(\log _{3}(x-1)=\log _{3} 4\).

Describe the change-of-base property and give an example.

By 2019 , nearly \(\$$ I out of every \)\$ 5\( spent in the U.S. economy is projected to go for health care. The bar graph shows the percentage of the U.S. gross domestic product \)(G D P)\( going toward health care from 2007 through \)2014,\( with a projection for 2019 The data can be modeled by the function \)f(x)=1.2 \ln x+15.7\( where \)f(x)\( is the percentage of the U.S. gross domestic product going toward health care \)x\( years after \)2006 .\( Use this information to solve. a. Use the function to determine the percentage of the U.S. gross domestic product that went toward health care in \)2008 .\( Round to the nearest tenth of a percent. Does this underestimate or overestimate the percent displayed by the graph? By how much? b. According to the model, when will \)18.6 \%$ of the U.S. gross domestic product go toward health care? Round to the nearest year.
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