/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 17 Solve each equation. $$ \log... [FREE SOLUTION] | 91影视

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Chapter 12: Problem 17

Solve each equation. $$ \log _{4} 2+\log _{4} x=0 $$

Short Answer

Expert verified
The solution is \( x = \frac{1}{2} \).

Step by step solution

01

Use the Logarithm Addition Property

We start with the equation: \( \log_{4} 2 + \log_{4} x = 0 \). By applying the logarithm addition property, \( \log_{a} b + \log_{a} c = \log_{a}(b \cdot c) \), we combine the logs: \( \log_{4} (2x) = 0 \).
02

Convert Logarithmic Equation to Exponential Form

Next, we need to convert the logarithmic equation \( \log_{4} (2x) = 0 \) into its exponential form. By definition, if \( \log_{a} b = c \), then \( a^c = b \). Therefore, \( 4^0 = 2x \).
03

Solve the Exponential Equation

We have the equation \( 4^0 = 2x \). Since \( 4^0 = 1 \), we have \( 1 = 2x \). Dividing both sides by 2 gives us \( x = \frac{1}{2} \).

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Key Concepts

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

Logarithm Addition Property
When dealing with logarithms, the addition property is a handy tool that helps simplify equations. If you encounter an equation like \( \log_{a} b + \log_{a} c \), you can combine the two logs by using the property that states:

  • \( \log_{a} b + \log_{a} c = \log_{a}(b \cdot c) \)
This means you can multiply the arguments (\(b\) and \(c\), in this case) into a single logarithm.

For example, in the exercise \( \log_{4} 2 + \log_{4} x = 0 \), you can merge the logs into:

\( \log_{4} (2x) = 0 \).

Knowing how to use the logarithm addition property allows you to simplify complex logarithmic expressions, making it easier to solve them.
Exponential Form
Converting a logarithmic equation to its exponential form is a crucial step in solving it. The logarithmic form \( \log_{a} b = c \) can be rewritten as an exponential equation:

  • \( a^c = b \)
This transformation is powerful because it turns the equation into a form that's often easier to manipulate and solve.

In our example, after using the logarithm addition property, we end up with the equation \( \log_{4} (2x) = 0 \). By converting into exponential form, we express this as:

\( 4^0 = 2x \).

Understanding exponential form is key, as it reveals the corresponding relationship between the components of the logarithm, thereby simplifying the path to find the solution.
Solving Equations
Once an equation is simplified and transformed, solving it becomes a straightforward process. For the exponential form \( 4^0 = 2x \), it helps to remember that any non-zero number raised to the power of zero is 1:

  • \( 4^0 = 1 \)
This step simplifies our equation to \( 1 = 2x \), making it easy to solve for \(x\).

To isolate \(x\), divide both sides of the equation by 2, resulting in:

  • \( x = \frac{1}{2} \)
Solving equations is about systematically reducing them to their simplest form, allowing you to uncover the unknown variable. Through steps like applying the addition property and converting to exponential form, you make the equations easier to handle.

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Most popular questions from this chapter

Graph each logarithmic function. $$ f(x)=\log _{1 / 2} x $$

The formula \(y=y_{0} e^{k t}\) gives the population size y of a population that experiences a relative growth rate \(k(k\) is positive if growth is increasing and \(k\) is negative if growth is decreasing). In this formula, \(t\) is time in years and \(y_{0}\) is the initial population at time \(0 .\) Use this formula to solve Exercises 55 and \(56 .\) Round answers to the nearest year. (Source for data: U.S. Census Bureau and Federal Reserve Bank of Chicago) In \(2009,\) the population of Michigan was approximately 9,970,000 and decreasing according to the formula \(y=y_{0} e^{-0.003 t}\). Assume that the population continues to decrease according to the given formula and predict how many years after which the population of Michigan will be \(9,500,000 .\) (Hint: Let \(y_{0}=9,970,000 ; y=9,500,000\), and solve for \(t\).)

Simplify. $$ \log _{9} 9 $$

Solve. $$ \log _{6} 6^{-2}=x $$

Find the value of each logarithmic expression. $$ \log _{8} \frac{1}{2} $$
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