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Chapter 12: Problem 28

Write in exponential form. \(\log _{10} 10^{-2}=-2\)

Short Answer

Expert verified
10^{-2} = 10^{-2}

Step by step solution

01

Identify the Logarithmic Equation

Recognize the given logarithmic equation: \ \ \( \log _{10} 10^{-2} = -2 \)
02

Understand the Relationship

Recall that \ \( \log_b(a) = c \) can be written in exponential form as \ \( b^c = a \). Here, \( b = 10 \), \ \( c = -2 \), and \ \( a = 10^{-2} \)
03

Convert to Exponential Form

Rewrite the equation in exponential form: \ \( 10^{-2} = 10^{-2} \)

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Key Concepts

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

Logarithmic Equations
Logarithmic equations involve logarithms, which are the inverse operations of exponentials. They help us understand how many times a certain base number must be multiplied by itself to get a certain value. In the equation \(\log_{10} 10^{-2} = -2\), the logarithm tells us that 10 raised to the power of -2 equals \(10^{-2}\). To solve logarithmic equations, you often need to convert them to exponential form. This allows you to manipulate the equation more easily and understand the relationship between the base, exponent, and value.
Exponential Equations
Exponential equations feature the variable as the exponent. They are written in the form \( b^x = a \), where \( b\) is the base, \( x \) is the exponent, and \( a \) is the result. For example, in the exercise \(\log_{10} 10^{-2} = -2\), we convert the logarithmic form to exponential form: \( 10^{-2} = 10^{-2}\). This conversion shows us directly that raising 10 to the power of -2 gives \(10^{-2}\). Understanding how to move between logarithmic and exponential forms is crucial in solving these equations.
Base and Exponent
The base and exponent are fundamental elements in both exponential and logarithmic equations. The base is the number that is being raised to a power, while the exponent is the power to which the base is raised. In the given exercise, \( \log_{10} 10^{-2} = -2 \), the base is 10 and the exponent is -2. Converting to exponential form, we restate the equation as \( 10^{-2} = 10^{-2}\). Recognizing these components helps to simplify and solve the equations more effectively. It's essential to remember that the exponent indicates how many times you multiply the base by itself, and a negative exponent means you take the reciprocal.

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Most popular questions from this chapter

Use the properties of logarithms to express each logarithm as a sum or difference of logarithms, or as a single logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{4} 6^{2}$$

Use the change-of-base rule (with either common or natural logarithms) to approximate each logarithm to four decimal places. \(\log _{1 / 3} 7\)

To four decimal places, the values of \(\log _{10} 2\) and \(\log _{10} 9\) are $$\log _{10} 2=0.3010 \text { and } \log _{10} 9=0.9542$$ Use these values and the properties of logarithms to evaluate each expression. DO NOT USE A CALCULATOR. $$\log _{10} 2^{19}$$

To four decimal places, the values of \(\log _{10} 2\) and \(\log _{10} 9\) are $$\log _{10} 2=0.3010 \text { and } \log _{10} 9=0.9542$$ Use these values and the properties of logarithms to evaluate each expression. DO NOT USE A CALCULATOR. $$\log _{10} \frac{1}{9}$$

Use the properties of logarithms to express each logarithm as a sum or difference of logarithms, or as a single logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{6} \sqrt{\frac{p q}{7}}$$
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