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Chapter 9: Problem 73

Write an equivalent exponential equation. $$ \log _{5} 125=3 $$

Short Answer

Expert verified
The equivalent exponential equation is \( 5^3 = 125 \).

Step by step solution

01

Understand the Logarithmic Equation

The given equation is \( \log_{5} 125 = 3 \), which means in logarithmic form that the base \( 5 \) raised to the power of \( 3 \) equals \( 125 \).
02

Convert the Logarithmic Equation to an Exponential Equation

To convert the logarithmic equation of the form \( \log_{b}(a) = n \) to its equivalent exponential form, you use \( b^n = a \). In this problem, substitute \( b = 5 \), \( n = 3 \), and \( a = 125 \).
03

Write the Exponential Equation

Using the conversion from Step 2, the exponential form of the equation is \( 5^3 = 125 \), which confirms that the base \( 5 \) raised to the power of \( 3 \) is indeed \( 125 \).

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Key Concepts

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

Logarithms
Logarithms are mathematical expressions that help us solve equations involving powers more easily. They are the inverse operation of exponents. When you have a logarithm like \( \log_{b}(a) = n \), the base \( b \) is the number that needs to be raised to the power \( n \) to get \( a \). In simpler terms, a logarithm answers the question "To what power should we raise this base to get that number?"
  • If we say \( \log_{5}(125) = 3 \), it means the power that base \( 5 \) needs to be raised to make \( 125 \) is 3.
  • Logarithms make it easier to handle large numbers or solve exponential equations.
By understanding logarithms, you can seamlessly switch between exponential forms and logarithmic forms.
Base and Exponent Concepts
Understanding the base and exponent in mathematical terms is key for solving logarithmic and exponential equations. The base, often denoted as \( b \), is the number that gets multiplied by itself, and the exponent, shown as \( n \), is how many times to multiply the base by itself. Consider the expression \( b^n \):
  • \( b \) is the base - it's like the foundation of a building.
  • \( n \) is the exponent - it tells you how many floors (multiplications) the building has.
For example, in the expression \( 5^3 \), 5 is the base, and 3 is the exponent, meaning you'll multiply 5 by itself three times: \( 5 \times 5 \times 5 = 125 \). This understanding is essential for converting between logarithmic and exponential forms.
Equivalent Equations
Equivalent equations are different ways of expressing the same mathematical relationship. In the context of logarithms and exponents, they help in understanding and solving equations by converting them into simpler or more familiar forms. When you have an equation like \( \log_{5}(125) = 3 \), you can convert it to an exponential equation, \( 5^3 = 125 \), to make it easier to understand or solve. This conversion doesn't change the value, but it offers a different perspective.
  • Logarithmic form: Expresses the equation using a logarithm, which can sometimes simplify solving it.
  • Exponential form: The same relationship presented as a multiplication of the base by itself numerous times, easier for some calculations.
By converting between these forms, you solidify your understanding of the relationship and develop flexibility in solving complex problems.

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Most popular questions from this chapter

For Exercises 15 and \(16,\) use the following information. Bacteria usually reproduce by a process known as binary fission. In this type of reproduction, one bacterium divides, forming two bacteria. Under ideal conditions, some bacteria reproduce every 20 minutes. Write the equation for modeling the exponential growth of this bacterium.

Colby and Elsu are solving \(\ln 4 x=5 .\) Who is correct? Explain your reasoning. Colby \(\begin{aligned} \ln 4 x &=5 \\ 10^{\ln } 4 x &=10^{5} \\ 4 x &=100,000 \\\ x &=25,000 \end{aligned}\) Elsu \(\begin{aligned} \ln 4 x &=5 \\ e^{\ln 4 x} &=e^{5} \\ 4 x &=e^{5} \\ x &=\frac{e^{5}}{4} \\ & \times 37.1033 \end{aligned}\)

For Exercises \(33-35,\) use the following information. A small corporation decides that 8\(\%\) of its profits would be divided among its six managers. There are two sales managers and four nonsales managers. Fifty percent would be split equally among all six managers. The other 50\(\%\) would be split among the four nonsales managers. Let \(p\) represent the profits. Write an expression to represent the share of the profits each nonsales manager will receive.

Solve each equation or inequality. Round to the nearest ten-thousandth. \(e^{5 x} \geq 25\)

ACT/SAT If \(2^{4}=3^{x}\) , then what is the approximate value of \(x ?\) $$ \begin{array}{l}{\text { A } 0.63} \\ {\text { B } 2.34} \\ {\text { C } 2.52} \\ {\text { D } 4}\end{array} $$
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