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Chapter 4: Problem 37

Simplify the expression. $$ \log _{5} 5 $$

Short Answer

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Step by step solution

01

Identify the Logarithm Base

The base of the logarithm in the expression is 5, which is indicated by the subscript.
02

Understand the Logarithm Definition

Recall that \(\text{log}_b(a)\) means the exponent to which the base \(b\) must be raised to produce the value \(a\). In this case, \(\text{log}_5(5)\) means the exponent to which 5 must be raised to produce 5.
03

Simplify the Expression

Since \(5^1 = 5\), the exponent needed is 1. Therefore, \(\text{log}_5(5) = 1\).

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Key Concepts

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

Logarithm Base
The logarithm base is a fundamental part of understanding logarithms. In the expression \(\text{log}_5 5\), the base is the small number found at the bottom right of 'log'.
Here it is 5.
The base determines the number you need to repeatedly multiply to reach another number, called the argument.
For example, in \(\text{log}_5 5\), the base is the number 5. It tells you that 5 is the number being used to find the exponent that results in 5.

Remember: the base is always a positive number and not equal to 1.
Exponents
To fully grasp logarithms, you need to understand exponents. An exponent indicates how many times a number is multiplied by itself.
For instance, in \[ 2^3 = 2 \times 2 \times 2 = 8 \], 3 is the exponent, and 2 is being multiplied by itself three times to get 8.
Logarithms are the 'inverse' of exponents. This means they undo what exponents do.

When you see \(\text{log}_b(a)\), you are finding the exponent that base b needs to be raised to get 'a'. If \(\text{log}_5 5\) = 1, this tells us that 5 raises to the power of 1 equals 5.
Logarithm Definition
The definition of a logarithm is crucial. A logarithm answers the question: 'To what power must the base be raised, to produce the argument?'
Using our example \(\text{log}_5 5\), we want to know how many times we need to multiply 5 by itself to get 5.
In this case, \(5^1 = 5\), so \(\text{log}_5 5 = 1\).
Here are a few helpful points to remember:
  • The base is the number being raised to a power
  • The exponent is the output of the logarithm function
  • The argument is the number you want to obtain by raising the base to the exponent

So, \(\text{log}_b(a) = c\) means that base b raised to the power of c equals a\(b^c = a\).

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Most popular questions from this chapter

A table of data is given. a. Graph the points and from visual inspection, select the model that would best fit the data. Choose from $$\begin{array}{ll} y=m x+b \text { (linear) } & y=a b^{x} \text { (exponential) } \\ y=a+b \ln x \text { (logarithmic) } & y=\frac{c}{1+a e^{-b x}} \text { (logistic) } \end{array}$$ b. Use a graphing utility to find a function that fits the data. $$ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 640 \\ \hline 20 & 530 \\ \hline 40 & 430 \\ \hline 50 & 360 \\ \hline 80 & 210 \\ \hline 100 & 90 \\ \hline \end{array} $$

Find the difference quotient \(\frac{f(x+h)-f(x)}{h} .\) Write the answers in factored form. $$f(x)=2^{x}$$

Compare the graphs of the functions. $$ Y_{1}=\ln (2 x) \quad \text { and } \quad Y_{2}=\ln 2+\ln x $$

Suppose that \(P\) dollars in principal is invested in an account earning \(3.2 \%\) interest compounded continuously. At the end of 3 yr, the amount in the account has earned \(\$ 806.07\) in interest. a. Find the original principal. Round to the nearest dollar. (Hint: Use the model \(A=P e^{r t}\) and substitute \(P+806.07\) for \(A .)\) b. Using the original principal from part (a) and the model \(A=P e^{r t},\) determine the time required for the investment to reach \(\$ 10,000\).

Solve the equation. Write the solution set with the exact values given in terms of common or natural logarithms. Also give approximate solutions to 4 decimal places. \(10^{5+8 x}+4200=84,000\)
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