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Chapter 9: Problem 46

Write each exponential equation as a logarithmic equation. See Example 2. $$ m^{n}=p $$

Short Answer

Expert verified
\( n = \log_{m}(p) \)

Step by step solution

01

Understanding the Exponential Equation

The given exponential equation is \( m^{n} = p \). In this equation, \( m \) is the base, \( n \) is the exponent, and \( p \) is the result.
02

Writing the Corresponding Logarithmic Form

To convert the exponential equation to its logarithmic form, use the relationship between exponents and logarithms. The general form is: if \( a^{b} = c \), then the logarithmic form is \( b = \log_{a}(c) \). Apply this to the given equation.
03

Applying the Formula

For the equation \( m^{n} = p \), the base \( m \) becomes the base of the logarithm, \( n \) is the output of the log equation, and \( p \) is the input to the logarithm. Therefore, the logarithmic form is \( n = \log_{m}(p) \).

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Key Concepts

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

Exponential Equations
Exponential equations are a fundamental aspect of algebra that involve variables in the exponent. In an exponential equation, such as \( m^n = p \):
  • \( m \) is the base, which is a constant value.
  • \( n \) is the exponent, indicating how many times the base is multiplied by itself.
  • \( p \) is the result of the exponentiation.
Exponential equations are used to model situations involving growth or decay, such as population growth, radioactive decay, or compound interest.
Understanding how these equations work lays the groundwork for converting them into other forms, like logarithmic equations, which can be more convenient for solving certain types of problems.
Converting Equations
Converting equations between different forms is an essential skill in algebra. Specifically, moving from exponential form to logarithmic form helps in analyzing and solving equations more easily. To convert an exponential equation like \( a^b = c \) into a logarithmic equation, we rely on the relationship between exponentials and logarithms.
  • The base of the exponential equation becomes the base of the logarithm.
  • The exponent is the equivalent of the result in the logarithmic form.
  • The result of the exponential equation becomes the input value for the logarithmic expression.
This conversion is based on the formula: if \( a^b = c \), then \( b = \log_a(c) \).
By converting equations, we can tackle problems using logarithmic properties, which often simplify solving for unknowns.
Logarithmic Form
Logarithms are mathematical functions that help determine the power to which a base must be raised to obtain a particular number. The logarithmic form is derived through conversion from an exponential equation, aiding in solving exponential problems.
For an equation like \( m^n = p \), once converted, we get \( n = \log_m(p) \). Here's what the components represent:
  • \( n \) is the exponent from the original equation, now the result of the logarithm.
  • \( m \), the base of the exponent, is also the base of the log.
  • \( p \) is the number we are "taking the log of" to find \( n \).
The logarithmic form exposes the inverse nature of exponents and logs, making it an invaluable tool for solving equations where the variable is an exponent.
Knowing how to write equations in logarithmic form facilitates understanding and solving complex algebraic expressions.

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Most popular questions from this chapter

Population Growth. \(\quad\) A population growing continuously at an annual rate \(r\) will triple in a time \(t\) given by the formula \(t=\frac{\ln 3}{r} .\) How long will it take the population of a town to triple if it is growing at the rate of \(12 \%\) per year?

Epidemics. The spread of hoof-and-mouth disease through a herd of cattle can be modeled by the function \(P(t)=2 e^{0.27 t}\) ( \(t\) is in days). If a rancher does not quickly treat the two cows that now have the disease, how many cattle will have the disease in 12 days?

Oceanography. The width \(w\) (in millimeters) of successive growth spirals of the sea shell Catapulus voluto, shown below, is given by the exponential function \(w(n)=1.54 e^{0.503 n}\) where \(n\) is the spiral number. Find the width, to the nearest tenth of a millimeter, of the sixth spiral.

Simplify each complex fraction. $$ \frac{2+\frac{1}{x^{2}-1}}{1+\frac{1}{x-1}} $$

Use the tables of values for functions \(f\) and \(g\) to find each of the following. a. \((f+g)(1)\) b. \((f-g)(5)\) c. \((f \cdot g)(1)\) d. \((g / f)(5)\) $$ \begin{array}{|c|c|c|c|} \hline x & f(x) & x & g(x) \\ \hline 1 & 3 & 1 & 4 \\ 5 & 8 & 5 & 0 \\ \hline \end{array} $$
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