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Chapter 4: Problem 62

Simplify the expression. $$ \log _{d} d $$

Short Answer

Expert verified
\( \log_d d = 1 \)

Step by step solution

01

Understand the Logarithm Definition

Recall that the logarithm \( \log_b (a) \) represents the exponent to which the base \( b \) must be raised to produce \( a \). In other words, \( \log_b (a) = c\) implies \( b^c = a \).
02

Apply the Logarithm Definition

For the given expression \( \log_d d \), identify that \( d \) is both the base and the argument. According to the logarithm definition: \((d^1 = d)\), thus \(\log_d d = 1\).
03

Write the Simplified Form

Conclude from the previous steps that the expression simplifies to 1.

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Key Concepts

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

logarithm definition
The logarithm is a fundamental mathematical concept used often in algebra and higher mathematics. To understand it, start with exponential notation. If you have an equation like b^c = a, it means that the base b raised to the exponent c results in a. Now, the logarithm asks the inverse question: what exponent do we need to raise b to get a? The notation for this is log_b (a), which reads as 'the log base b of a'. It essentially reflects the exponent needed to reach from the base to the given number.

For example:
  • If 2^3 = 8, then log_2 (8) = 3.
  • If 10^2 = 100, then log_{10} (100) = 2.

Understanding this definition is key to solving problems involving logarithms.
logarithmic simplification
Simplifying logarithmic expressions often involves using various properties of logarithms. In the case of the exercise log_d (d), an important property to use is that the log of a number n to its own base b equals 1. This is derived from the definition that b^1 = b.

Here's a step-by-step example using this property:
  • Identify the base d and the argument d in log_d (d).
  • Recognize that any number raised to the power of 1 remains unchanged. So, d^1 = d.
  • Because of this, log_d (d) = 1. This simplification leverages the logarithmic property we stated.
This property significantly eases the process of dealing with logarithmic equations involving identical base and argument.
base and argument in logarithms
In logarithmic expressions, two key components to consider are the 'base' and the 'argument'.
  • Base: The base is the number that is raised to a power. It is the small subscript number in the logarithmic notation. For instance, in log_2 (8), the base is 2.
  • Argument: The argument is the number we are taking the logarithm of. It is located next to the base in the notation. In the same example log_2 (8), the argument is 8.

Understanding the role of these two elements is crucial:
  • A high base and low argument typically translate to small logarithmic values (exponents).
  • If the base and argument are the same, like in log_d (d), the answer or logarithmic value is always 1.
Being clear on these roles helps in the correct application of logarithmic definitions and properties during problem-solving.

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Most popular questions from this chapter

Determine if the statement is true or false. For each false statement, provide a counterexample. For example, \(\log (x+y) \neq \log x+\log y\) because \(\log (2+8) \neq \log 2+\log 8\) (the left side is 1 and the right side is approximately 1.204 ). $$ \ln 10=\frac{1}{\log e} $$

a. Write the difference quotient for \(f(x)=\ln x\). b. Show that the difference quotient from part (a) can be written as \(\ln \left(\frac{x+h}{x}\right)^{1 / h}\).

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. \(4 \ln (6-5 t)+2=22\)

Compare the graphs of \(Y_{1}=\frac{e^{x}-e^{-x}}{2}\), \(\mathrm{Y}_{2}=\ln \left(x+\sqrt{x^{2}+1}\right)\), and \(\mathrm{Y}_{3}=x\) on the viewing window [-15.1,15.1,1] by \([-10,10,1] .\) Based on the graphs, how do you suspect that the functions are related?

A function of the form \(P(t)=a b^{t}\) represents the population of the given country \(t\) years after January 1,2000 . a. Write an equivalent function using base \(e\); that is, write a function of the form \(P(t)=P_{0} e^{k t} .\) Also, determine the population of each country for the year 2000 . $$\begin{array}{|l|c|c|c|} \hline \text { Country } & P(t)=a b^{t} & P(t)=P_{0} e^{k t} & \begin{array}{c} \text { Population } \\ \text { in } 2000 \end{array} \\ \hline \text { Haiti } & P(t)=8.5(1.0158)^{t} & & \\ \hline \text { Sweden } & P(t)=9.0(1.0048)^{t} & & \\ \hline \end{array}$$ b. The population of the two given countries is very close for the year 2000 , but their growth rates are different. Determine the year during which the population of each country will reach 10.5 million. c. Haiti had fewer people in the year 2000 than Sweden. Why did Haiti reach a population of 10.5 million sooner?
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