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Chapter 4: Problem 27

Write the equation in logarithmic form. $$ 5^{3}=125 $$

Short Answer

Expert verified
\( \log_{5}(125)=3 \)

Step by step solution

01

Understand the Exponential Form

The given equation is in exponential form: \[5^{3}=125\]This means that the base is 5, the exponent is 3, and the result is 125.
02

Identify the Components for Logarithmic Form

Recognize that the base in the exponential form becomes the base of the logarithm, the result of the exponential form becomes the argument of the logarithm, and the exponent becomes the value of the logarithm.
03

Rewrite in Logarithmic Form

Rewrite the equation in logarithmic form. The logarithmic form of \[5^{3}=125\]is: \[\log_{5}(125)=3\]

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Key Concepts

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

Exponential Form
Exponential form is a way of expressing repeated multiplication of the same number. In the exercise, we have the equation \(5^{3}=125\). Here, 5 is multiplied by itself three times to give 125. This format is generally written as \(b^{n}=a\), where 'b' is the base, 'n' is the exponent, and 'a' is the result. Exponential form is helpful for showing large numbers compactly. For example, rather than writing 5 * 5 * 5, we use \(5^{3}\). This makes understanding mathematical relationships clearer and more concise.
Logarithmic Form
Logarithmic form is essentially the inverse of exponential form. It answers the question: 'to what exponent must the base be raised, to produce a given number?'. From our exercise, the exponential form \(5^{3}=125\) can be rewritten in logarithmic form as \(\log_{5}(125)=3\)\. This means that 5 must be raised to the power of 3 to get 125. The general form is \(\log_{b}(a) =n\)\, where 'b' is the base, 'a' is the argument, and 'n' is the exponent. This form is very useful in solving different kinds of equations and understanding growth rates in both mathematics and real-world applications.
Base and Exponent
The 'base' and 'exponent' are key components in both exponential and logarithmic forms. In \(5^{3}=125\), the number 5 is the base. It's the number that is repeatedly multiplied. The 'exponent' is the 3. It's the number of times the base is multiplied by itself. Understanding these parts is crucial because they help convert between exponential and logarithmic forms. For instance, in logarithmic notation \(\log_{5}(125)=3\)\, 5 is still the base and 3 represents the exponent from the original exponential equation.
Logarithm Properties
Logarithms have several properties that make calculations easier. One useful property is the 'Product Property', \( \log_{b}(xy) = \log_{b}(x) + \log_{b}(y)\)\. This helps in breaking down complex multiplication into simpler additions. Another is the 'Quotient Property', \( \log_{b}(\frac{x}{y}) = \log_{b}(x) - \log_{b}(y)\)\, for simplifying divisions. The 'Power Property' states \( \log_{b}(x^{n}) = n\log_{b}(x)\)\, to easily manage exponents. Understanding these properties can greatly simplify solving logarithmic equations and make handling large numbers more manageable.

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Most popular questions from this chapter

Determine if the statement is true or false. For each false statement, provide a counterexample. For example, \(\log (x+y) \neq \log x+\log y\) because \(\log (2+8) \neq \log 2+\log 8\) (the left side is 1 and the right side is approximately 1.204 ). $$ \log _{8}\left(\frac{1}{w}\right)=-\log _{8} w $$

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. \(2^{1-6 x}=7^{3 x+4}\)

Show that \(-\ln \left(x-\sqrt{x^{2}-1}\right)=\ln \left(x+\sqrt{x^{2}-1}\right)\).

(See Example 8 ) a. Estimate the value of the logarithm between two consecutive integers. For example, \(\log _{2} 7\) is between 2 and 3 because \(2^{2}<7<2^{3}\). b. Use the change-of-base formula and a calculator to approximate the logarithm to 4 decimal places. c. Check the result by using the related exponential form. $$ \log _{5} 3 $$

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. \(1024=19^{x}+4\)
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