Problem 16
Question
Evaluate the logarithm using the change-of-base formula. Round your result to three decimal places.$$\log _{1 / 8} 64$$.
Step-by-Step Solution
Verified Answer
The expression \(\log _{1 / 8} 64\) evaluates approximately to -2.000 when rounded to three decimal places.
1Step 1: Apply the Change of Base Formula
To evaluate the logarithm we use the change-of-base formula. The formula used to evaluate logarithms in a different base than given is \(\log_b a = \log_c a / \log_c b\). So, \(\log _{1 / 8} 64 = \log_{10} 64 / \log_{10} (1/8)\).
2Step 2: Evaluate the Logarithms
Calculate the value of log 64 and log (1/8) to the base 10. Log 64 to the base 10 is approximately 1.806 and log (1/8) to the base 10 is -0.903. So, now we have \(\log _{1 / 8} 64 = 1.806/-0.903\).
3Step 3: Perform the Division
Perform the division to get the final answer. So, \(\log _{1 / 8} 64 = 1.806/-0.903 = -2.000 \).
Key Concepts
LogarithmsEvaluating LogarithmsLogarithmic Functions
Logarithms
To grasp the concept of a logarithm, imagine it as a way to ask the following question: 'To what power must we raise a given base to get a certain number?' In mathematical terms, if we have a base \( b \) and a number \( x \), and we want to find the power \( y \) such that \( b^y = x \), then \( y \) is the logarithm of \( x \) to base \( b \), which is notated as \( \log_b(x) = y \).
Logarithms are fundamental in various mathematical applications, often simplifying multiplication and division into addition and subtraction, which, in the context of algebra and calculus, can make problems more manageable. For example, the logarithmic identities like \( \log(ab) = \log(a) + \log(b) \) and \( \log(a/b) = \log(a) - \log(b) \) are enormously helpful.
Understanding logarithms is a stepping stone to mastering exponential functions, as they are essentially the inverse of exponentiation. In the real world, logarithms help us model phenomena that grow exponentially, like interest in finance or population growth in biology.
Logarithms are fundamental in various mathematical applications, often simplifying multiplication and division into addition and subtraction, which, in the context of algebra and calculus, can make problems more manageable. For example, the logarithmic identities like \( \log(ab) = \log(a) + \log(b) \) and \( \log(a/b) = \log(a) - \log(b) \) are enormously helpful.
Understanding logarithms is a stepping stone to mastering exponential functions, as they are essentially the inverse of exponentiation. In the real world, logarithms help us model phenomena that grow exponentially, like interest in finance or population growth in biology.
Evaluating Logarithms
Evaluating logarithms is the process of finding the numeric value of a logarithm. To do this effectively, you might need to understand a few properties of logarithms. One of the key approaches to evaluating complex logarithms is through the lens of the change-of-base formula.
That formula, as shown in the original exercise, is \[ \log_b a = \frac{\log_c a}{\log_c b} \], where \( c \) is any positive value, commonly 10 or \( e \) (the base of natural logarithms). Why does this matter? It allows you to use a standard scientific calculator to evaluate logarithms with bases that aren't immediately available by transforming them into a base that the calculator can handle.
Evaluating logarithms, particularly with a calculator, typically involves rounding. However, this should be deferred to the last step to minimize rounding errors. Furthermore, appreciating logarithmic properties can sometimes let you evaluate a logarithm without a calculator entirely. For instance, \( \log_2 8 \) is clearly 3 because \( 2^3 = 8 \), following the essential definition of logarithms.
That formula, as shown in the original exercise, is \[ \log_b a = \frac{\log_c a}{\log_c b} \], where \( c \) is any positive value, commonly 10 or \( e \) (the base of natural logarithms). Why does this matter? It allows you to use a standard scientific calculator to evaluate logarithms with bases that aren't immediately available by transforming them into a base that the calculator can handle.
Evaluating logarithms, particularly with a calculator, typically involves rounding. However, this should be deferred to the last step to minimize rounding errors. Furthermore, appreciating logarithmic properties can sometimes let you evaluate a logarithm without a calculator entirely. For instance, \( \log_2 8 \) is clearly 3 because \( 2^3 = 8 \), following the essential definition of logarithms.
Logarithmic Functions
A logarithmic function is, simply put, a function that involves a logarithm. More formally, it's written as \( f(x) = \log_b(x) \), where \( b \) is the base, and the domain of the function is all positive real numbers. Like their counterpart–exponential functions–logarithmic functions have their unique characteristics. They pass the horizontal line test, which implies they are one-to-one and thus have inverses.
Graphically, logarithmic functions have a vertical asymptote on the y-axis, and they increase or decrease based on the base \( b \). If \( b > 1 \), the function increases; if \( 0 < b < 1 \), it decreases. Moreover, they display notable behavior on a graph, curving gently away from the vertical asymptote but never touching or crossing it.
Understanding logarithmic functions is critical for solving equations where the variable is in an exponent, which is common in exponential growth and decay problems, as well as in the field of sound intensity measured in decibels, Richter scale for measuring earthquake intensities, and in pH levels that measure acidity in chemistry.
Graphically, logarithmic functions have a vertical asymptote on the y-axis, and they increase or decrease based on the base \( b \). If \( b > 1 \), the function increases; if \( 0 < b < 1 \), it decreases. Moreover, they display notable behavior on a graph, curving gently away from the vertical asymptote but never touching or crossing it.
Understanding logarithmic functions is critical for solving equations where the variable is in an exponent, which is common in exponential growth and decay problems, as well as in the field of sound intensity measured in decibels, Richter scale for measuring earthquake intensities, and in pH levels that measure acidity in chemistry.
Other exercises in this chapter
Problem 16
Use a graphing utility to create a scatter plot of the data. Decide whether the data could best be modeled by a linear model, an exponential model, or a logarit
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