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Chapter 6: Problem 1

A logarithm with base 10 is called a(n)________ logarithm.

Short Answer

Expert verified
A logarithm with base 10 is called a common logarithm.

Step by step solution

01

Identify the type of logarithm

Here, look at the description provided in the question. The question mentions a logarithm with base 10. The task is to identify the name of this type of logarithm.
02

Apply knowledge of logarithm terminology

In mathematics, the terminology used for a logarithm changes depending on the base. The base 10 logarithm has a specific name, which is a common logarithm.

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Key Concepts

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

Logarithm
A logarithm is a mathematical operation that answers the question, 'To what exponent must a fixed base number be raised to produce a given number?' In simpler terms, if you have an equation like \( b^x = y \), then the logarithm of \( y \) with base \( b \) is \( x \). It's represented as \( \log_b(y) = x \).

Understanding logarithms is crucial for solving exponential equations, where variables are in exponents. This concept is also a stepping stone to further topics in algebra, calculus, and beyond, as it features in many mathematical models and real-world applications such as in calculating interest rates or measuring sound intensity.
Base 10 Logarithm
Now, specifically focusing on the base 10 logarithm, which is famously referred to as the 'common logarithm,' mathematicians use the notation \( \log(y) \) without explicitly mentioning the base. This is due to its widespread usage, especially in scientific calculations where the base 10 is extremely handy because of our decimal number system. For example, if you see \( \log(100) = x \), you're really being asked 'To which power must 10 be raised to get 100?' And since \( 10^2 = 100 \), \( x \) would be 2.
Math Terminology
Understanding math terminology is essential for clear communication of complex ideas. In the case of the 'common logarithm,' it's a term that specifically denotes the base 10 logarithm, which is a convention agreed upon by mathematicians. Each field within mathematics 鈥� whether it be algebra, geometry, or statistics 鈥� has its own specific terminology. Getting familiar with these terms helps you to be more efficient and accurate in both learning new concepts and solving mathematical problems.
Algebra 2
Algebra 2 is a stage in math education where students dive deeper into the world of functions, expressions, and complex equations. It introduces new aspects of algebra, including logarithms, that build upon the foundations laid in earlier classes. Common logarithms, among other types, are extensively used throughout Algebra 2, particularly when dealing with exponential growth and decay, or when solving for variables in the exponent. A strong grasp of logarithms enables students to tackle real-world problems more effectively, as they can reverse engineer the process of exponentiation.

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Most popular questions from this chapter

In Exercises 13 and 14, describe and correct the error in simplifying the expression. $$ \begin{aligned} \left(4 e^{3 x}\right)^2 &=4 e^{(3 x)(2)} \\ &=4 e^{6 x} \end{aligned} $$

Explain why \(A=P\left(1+\frac{r}{n}\right)^{n t}\) approximates \(A=P e^{r t}\) as \(n\) approaches positive infinity.

When X-rays of a fixed wavelength strike a material \(x\) centimeters thick, the intensity \(I(x)\) of the X-rays transmitted through the material is given by \(I(x)=I_0 e^{-\mu x}\), where \(I_0\) is the initial intensity and \(\mu\) is a value that depends on the type of material and the wavelength of the X-rays. The table shows the values of \(\mu\) for various materials and X-rays of medium wavelength. $$ \begin{array}{|l|c|c|c|} \hline \text { Material } & \text { Aluminum } & \text { Copper } & \text { Lead } \\ \hline \text { Value of } \mu & 0.43 & 3.2 & 43 \\ \hline \end{array} $$a. Find the thickness of aluminum shielding that reduces the intensity of \(\mathrm{X}\)-rays to \(30 \%\) of their initial intensity. (Hint: Find the value of \(x\) for which \(I(x)=0.3 I_0\). b. Repeat part (a) for the copper shielding. c. Repeat part (a) for the lead shielding. d. Your dentist puts a lead apron on you before taking X-rays of your teeth to protect you from harmful radiation. Based on your results from parts (a)-(c), explain why lead is a better material to use than aluminum or copper.

Identify the focus, directrix, and axis of symmetry of the parabola. Then graph the equation. \(y=4 x^2\)

Solve the equation.\(5^{2 x}+20 \cdot 5^x-125=0\)
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