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Chapter 4: Problem 14

\(\log _{b} b=\) _____ because \(b^{\square}=b\).

Short Answer

Expert verified
\text{log}_b b = 1 because } b^1 = b \text{.

Step by step solution

01

Understand the Logarithm

The logarithm \(\text{log}_b b\) is the power to which the base \(b\) must be raised to produce itself.
02

Define the Exponential Form

Recall that \( \text{log}_b (a) = x \) means \( b^x = a \). In this case, set \(a = b\): \( \text{log}_b (b) = x \)
03

Match the Exponent

Since \( b^x = b \), it becomes clear that \( x \) must be 1 because any number raised to the power 1 is itself.
04

Write the Solution

Therefore, \( \text{log}_b b = 1 \).

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Key Concepts

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

logarithmic properties
Logarithms are mathematical operations related to exponents. They are useful for solving equations involving exponential growth or decay. A logarithm tells us the power to which a base number is raised to get another number.
exponential form
Every logarithmic expression can be rewritten in its exponential form, which can sometimes make it easier to understand or solve. For instance, the logarithmic equation \(\text{log}_b (a) = x\) means \(b^x = a\). This is fundamental because it shows the direct relationship between logarithms and exponents. In our exercise, \(\text{log}_b (b) = 1\) because \(b^1 = b\). Remembering this conversion is handy for various mathematical and practical applications.
base of logarithm
The base of a logarithm is the number that is repeatedly multiplied to reach another number. In the expression \(\text{log}_b (a)\), \(b\) is the base. For our specific problem, where \(\text{log}_b (b)\) calculates to 1, it highlights a basic yet important property: the logarithm of any number to itself is always 1. This matters because it provides a foundation for more advanced logarithmic and exponential operations, reinforcing the unity and identity properties in algebra.

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Most popular questions from this chapter

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. \(10^{3+4 x}-8100=120,000\)

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. \(\log _{5} z=3-\log _{5}(z-20)\)

Use the model \(A=P e^{r t} .\) The variable \(A\) represents the future value of \(P\) dollars invested at an interest rate \(r\) compounded continuously for \(t\) years. If a couple has \(\$ 80,000\) in a retirement account, how long will it take the money to grow to \(\$ 1,000,000\) if it grows by \(6 \%\) compounded continuously? Round to the nearest year.

a. Graph \(f(x)=\ln x\) and \(g(x)=(x-1)-\frac{(x-1)^{2}}{2}+\frac{(x-1)^{3}}{3}-\frac{(x-1)^{4}}{4}\) on the viewing window [-2,4,1] by [-5,2,1] . How do the graphs compare on the interval (0,2) ? b. Use function \(g\) to approximate \(\ln 1.5\). Round to 4 decimal places.

The number of computers \(N(t)\) (in millions) infected by a computer virus can be approximated by $$N(t)=\frac{2.4}{1+15 e^{-0.72 t}}$$ where \(t\) is the time in months after the virus was first detected. a. Determine the number of computers initially infected when the virus was first detected. b. How many computers were infected after 6 months? Round to the nearest hundred thousand. c. Determine the amount of time required after initial detection for the virus to affect 1 million computers. Round to 1 decimal place. d. What is the limiting value of the number of computers infected according to this model?
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