Question

Write the following using indices: (a) \(\log _{5} 625=4\) (b) \(\log _{2} 256=8\) (c) \(\log 0.0251=-1.6\) (d) \(\ln 17=2.8332\)

Short answer

Question: Rewrite the following logarithmic expressions in index notation. (a) \(\log _{5} 625=4\) (b) \(\log _{2} 256=8\) (c) \(\log 0.0251=-1.6\) (d) \(\ln 17=2.8332\) Answer: (a) \(5^4 = 625\) (b) \(2^8 = 256\) (c) \(10^{-1.6} = 0.0251\) (d) \(e^{2.8332} = 17\)

Step-by-step solution

(a) Rewrite \(\log _{5} 625=4\) in index notation

Using the logarithmic property, let's rewrite the equation as: \(5^4 = 625\)

(b) Rewrite \(\log _{2} 256=8\) in index notation

Using the logarithmic property, let's rewrite the equation as: \(2^8 = 256\)

(c) Rewrite \(\log 0.0251=-1.6\) in index notation

Since this is a common logarithm, we have \(\log x = \log_{10}x\). Therefore, let's rewrite the equation using the logarithmic property: \(10^{-1.6} = 0.0251\)

(d) Rewrite \(\ln 17=2.8332\) in index notation

Since this is a natural logarithm, it is understood as an expression with base \(e\). So we have \(\ln x = \log_{e}x\). Let's rewrite the equation as: \(e^{2.8332} = 17\)

Key concepts

Index Notation

Index notation is a convenient way to express large or small numbers in terms of powers or exponents. Understanding index notation is particularly useful when working with logarithms, as it allows us to simplify expressions and equations. For instance, when you see \(\log_{5} 625=4\), it can be rewritten in index notation as \(5^4 = 625\). Here’s how index notation works:

• The base (in the example above, 5) is the number that is multiplied by itself.
• The exponent (or index) shows how many times the base is used as a factor (in this case, 4).
• The outcome (625) is the value you get from multiplying the base by itself, as indicated.

Using this approach, any logarithmic equation can be converted to index notation, clarifying the relationship between its elements.

Logarithmic Property

Logarithmic properties form the backbone of working with logarithms and help to transform complex logarithmic equations into more manageable forms. One primary property is that logarithms allow us to translate between exponential and logarithmic forms.For example, if you have a logarithmic equation \(\log_{b} a = c\), it can be rewritten using the exponential form as \(b^c = a\). This is the fundamental technique applied in the textbook solution where we derive index notation from logarithmic expressions, like \(\log_{2} 256=8\) to \(2^8 = 256\).Understanding these transformations involves recognizing:

• The logarithm answers the question: "To what exponent must the base be raised, to produce this number?"
• Logarithmic equations can be rearranged to solve for unknown variables when rewritten in exponential form.
• Using properties of exponents helps simplify operations in logarithms, such as multiplying, dividing, and changing the base of the logarithm.

Natural Logarithm

The natural logarithm is a special type of logarithm where the base is the mathematical constant \(e\), approximately equal to 2.71828. This logarithm is widely used in mathematics, especially in calculus and other sciences due to its natural properties. It's often denoted as \(\ln x\), meaning \(\log_{e} x\).In the provided solution, for example, \(\ln 17=2.8332\) is converted to exponential form, resulting in \(e^{2.8332} = 17\). This illustrates how natural logarithms can be used to solve and express real-world problems where exponential growth and decay are present.Why use natural logarithms?

• Natural logarithms have properties that make them easier to work with in calculus.
• They are used in calculating half-life or time constants in physics and chemistry.
• It's the inverse function of the exponential function \(e^x\), making it crucial for integration and differentiation.

Common Logarithm

A common logarithm is a logarithm with base 10, symbolized simply as \(\log x\) without a base visible, indicating the base is implicitly 10. It's widely used in various fields due to the base being a factor of our decimal number system.In the textbook solution, the expression \(\log 0.0251=-1.6\) illustrates the use of common logarithms. This converts to \(10^{-1.6} = 0.0251\) in index notation, providing clarity and simplicity to solving logarithmic problems involving powers of ten.Why the focus on base 10?

• Base 10 is intuitive because it aligns with our natural way of counting and perceiving numbers.
• Common logarithms simplify calculations in engineering, science, and technology, largely when dealing with multiplicative factors of ten.
• They provide a straightforward approach to estimating large number scale differences, such as in measuring sound intensity in decibels.

Understanding these concepts will give you greater ease in applying logarithms to a variety of mathematical and practical contexts.