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Chapter 5: Problem 56

Given that \(\log _{a} 2=0.301, \log _{a} 7=0.845,\) and \(\log _{a} 11=1.041,\) find each of the following, if possible. Round the answer to the nearest thousandth. $$\log _{a} \frac{1}{7}$$

Short Answer

Expert verified
-0.845.

Step by step solution

01

Recall the Logarithm Property

Use the property of logarithms that states ' content='Use the property of logarithms that states -1) multiplying the given logarithm value by -1. Multiply the value of -0.845).
02

Apply the Known Value

Given: The value of - - - - - -0.845.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithm Rules
Understanding logarithm rules is crucial for solving many problems in math.
Let's delve deeper into some key logarithm rules you鈥檒l often come across:
  • Product Rule: \(\log_b(MN) = \log_b(M) + \log_b(N)\)
    When multiplying numbers inside a logarithm, you can split it into the sum of two logs.
  • Quotient Rule: \(\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)\)
    For division, it works similarly, but now it's the difference between the logs. In the exercise, this rule helps us to find \(\log_a\left(\frac{1}{7}\right)\).
  • Power Rule: \(\log_b(M^p) = p \cdot \log_b(M)\)
    When a number inside a log is raised to a power, you can bring the exponent down in front as a multiplier.
  • Change of Base Formula: \(\log_b(M) = \frac{\log_k(M)}{\log_k(b)}\)
    This formula lets you convert logarithms from one base to another, using a common base like 10 or e (natural logarithm).
Knowing these rules makes manipulating logarithms a lot easier and aids in simplifying complex problems.
Practice using each rule until they become second nature.
Change of Base Formula
The change of base formula is your friend when dealing with unfamiliar bases:
It states: \(\log_b(M) = \frac{\log_k(M)}{\log_k(b)}\)
  • You can choose any base 'k' for this conversion, although base 10 or e (natural logarithm) are the most common.
  • This formula is particularly useful when your calculator doesn鈥檛 support logarithms in the given base.
For example, if we need to find \(\log_2(8)\), and our calculator only handles base 10 or e, we can convert it:
\(\log_2(8) = \frac{\log(8)}{\log(2)} = \frac{\log_{10}(8)}{\log_{10}(2)}\)
Both the numerator and the denominator are now in base 10, which most calculators can handle.
Similarly, if you prefer natural logs: \(\log_2(8) = \frac{\ln(8)}{\ln(2)}\).
By mastering the change of base formula, you gain the flexibility to tackle any logarithm problem, regardless of the base.
Inverse Logarithm
Logarithms and exponents are inverses of each other, much like multiplication and division.
Understanding this relationship is key:
  • If you have \(y = \log_b(x)\), then \(b^y = x\). The logarithm answers the question: 'To what power must we raise the base b to get x?'
  • In other words, \(\log_b(x)\) undoes \(b^x\), and vice versa.
Let's look at an example: If \(\log_3(81) = y\), it means 3 raised to what power gives us 81?
Since \(3^4 = 81\), we can write \(\log_3(81) = 4\).
Conversely, if \(y = 4\), then \(3^y = 81\).
Recognizing logarithms as the inverse of exponentiation helps simplify many problems, including solving for unknowns and understanding the behavior of functions.
Remember: master the inverse nature of logarithms to gain confidence in manipulating and solving logarithmic equations.

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Most popular questions from this chapter

Price of Admission to the Magic Kingdom. In \(2015,\) the price of a one-day, one-park admission to Disney's Magic Kingdom in Florida rose to \(\$ 105 .\) The exponential function $$ D(x)=4.532(1.078)^{x} $$ where \(x\) is the number of years after \(1971,\) models the price of a ticket. (Source: AllEars.net, an independent Disney consumer website) Find the price of a ticket in \(1980,\) in \(2000,\) and in \(2012 .\) Then use the function to project the price of a ticket in 2020 .

Determine whether each of the following is true or false. Assume that \(a, x, M,\) and \(N\) are positive. $$\frac{\log _{a} M}{x}=\log _{a} M^{1 / x}$$

Growth of a Stock. The value of a stock is given by the function $$ V(t)=58\left(1-e^{-1.1 t}\right)+20 $$ where \(V\) is the value of the stock after time \(t,\) in months. a) Graph the function. b) Find \(V(1), V(2), V(4), V(6),\) and \(V(12)\) c) After how long will the value of the stock be \(\$ 75 ?\)

A function that will convert women's shoe sizes in the United States to those in Australia is $$ s(x)=\frac{2 x-3}{2} $$ (Source: OnlineConversion.com). a) Determine the women's shoe sizes in Australia that correspond to sizes \(5,7 \frac{1}{2},\) and 8 in the United States. b) Find a formula for the inverse of the function and explain what it represents. c) Use the inverse function to determine the women's shoe sizes in the United States that correspond to sizes \(3,5 \frac{1}{2},\) and 7 in Australia.

Interest in a College Trust Fund. Following the birth of his child, Benjamin deposits \(\$ 10,000\) in a college trust fund where interest is \(3.9 \%,\) compounded semiannually. a) Find a function for the amount in the account after \(t\) years. b) Find the amount of money in the account at \(t=0,4\) \(8,10,18,\) and 21 years.
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