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Chapter 4: Problem 141

For the following exercises, rewrite each equation in logarithmic form. $$19^{x}=y$$

Short Answer

Expert verified
\(x = \log_{19} y\)

Step by step solution

01

Identify the Exponential Form

The given equation is expressed in exponential form as \(19^x = y\). In an exponential equation, the base is \(19\), the exponent is \(x\), and the result is \(y\).
02

Apply the Definition of Logarithms

To rewrite the equation in logarithmic form, we need to understand the equivalence between exponents and logarithms. The definition of a logarithm states that if \(a^b = c\), then it can be expressed in logarithmic form as \(b = \log_a c\).
03

Rewrite Using Logarithmic Form

Using the definition of logarithms, rewrite the exponential equation \(19^x = y\) as \(x = \log_{19} y\). This expression indicates that \(x\) is the power to which \(19\) must be raised to obtain \(y\).

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Key Concepts

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

Exponential Form
When discussing mathematical expressions, exponential form is a way of representing repeated multiplication of a number by itself. In this form, you have a base number and an exponent.
For instance, in the equation \(19^x = y\):
  • The **base** is \(19\).
  • The **exponent** is \(x\).
  • The **result** is \(y\).
This notation is essential because it allows us to express large numbers compactly and perform operations on them more efficiently. For example, if you were to compute \(19^3\), you would multiply 19 by itself three times, yielding: \[19 \times 19 \times 19 = 6859\]The exponential form gives us a structured way to interpret and handle these expressions.
Base of Exponential
The base of an exponential expression serves as the foundation of repeated multiplication. It is the number that is multiplied by itself according to the value of the exponent.
For example, in \(19^x = y\), the base is 19. Recognizing the base is crucial when manipulating exponential expressions or converting them to other forms like logarithmic form. The importance of the base includes:
  • Determining the rate of growth or decay in exponential equations.
  • Setting the context for logarithmic conversions.
  • Establishing a pattern for calculations and predictions in fields such as finance, science, and engineering.
Understanding the base helps us interpret the behavior of exponential functions in real-world applications, from calculating interest to predicting population growth.
Definition of Logarithms
Logarithms are the inverse operations of exponentiation. They help us find the exponent to which a base must be raised to produce a certain number.
Understanding this is vital for converting exponential forms to logarithmic forms.
According to the definition of a logarithm, if \[a^b = c\]then it can be represented in logarithmic form as \[b = \log_a c\]In our case with \(19^x = y\), the logarithmic form becomes:\[x = \log_{19} y\]A logarithm answers the question "To which power must the base be raised to get the given number?"
  • They provide a method to simplify complex multiplication processes into addition, which is easier to perform.
  • They can transform exponential growth problems into linear problems, making them easier to analyze and solve.
Logarithms thus play a crucial role in simplifying calculations and solving equations that involve exponentials.

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Most popular questions from this chapter

Use like bases to solve the exponential equation. \(625 \cdot 5^{3 x+3}=125\)

For the following exercises, condense each expression to a single logaritim using the properties of logaritims. $$ \log (x)-\frac{1}{2} \log (y)+3 \log (z) $$

For the following exercises, use properties of logarithms to evaluate without using a calculator. $$ 2 \log _{9}(3)-4 \log _{9}(3)+\log _{9}\left(\frac{1}{729}\right) $$

Use the definition of a logarithm to rewrite the equation as an exponential equation. \(\log \left(\frac{1}{100}\right)=-2\)

For the following exercises, refer to Table 4.27. $$\begin{array}{|c|c|c|c|c|c|c|}\hline x & {1} & {2} & {3} & {4} & {5} & {6} \\\ \hline f(x) & {555} & {383} & {307} & {210} & {158} & {122} \\\ \hline\end{array}$$ Write the exponential function as an exponential equation with base \(e\).
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