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Chapter 4: Problem 330

Use the one-to-one property of logarithms to solve. \(\log _{4}(6-m)=\log _{4} 3 m\)

Short Answer

Expert verified
\( m = \frac{3}{2} \).

Step by step solution

01

Apply One-to-One Property

The one-to-one property of logarithms states that if \( \log_b (M) = \log_b (N) \), then \( M = N \). Therefore, from \( \log_{4}(6-m) = \log_{4} 3m \), we can write: \[ 6 - m = 3m \].
02

Solve for m

Starting with the equation from Step 1, \( 6 - m = 3m \), add \( m \) to both sides to get: \[ 6 = 4m \] Now, divide both sides by 4 to solve for \( m \): \[ m = \frac{6}{4} \], which simplifies to \[ m = \frac{3}{2} \].

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Key Concepts

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

One-to-One Property
In logarithmic equations, a useful property is the one-to-one property. This property allows us to equate the arguments of two logarithms directly, given that their bases are the same. If you have
  • \( \log_b(M) = \log_b(N) \)
then you can confidently write
  • \( M = N \)
This means that the expressions inside the logarithms are equal to each other. This property is incredibly helpful for simplifying and solving logarithmic equations. For example, consider
  • \( \log_4(6-m) = \log_4(3m) \)
Using the one-to-one property, we can directly conclude
  • \( 6-m = 3m \)
This step simplifies the equation significantly and allows us to focus on solving a basic algebraic equation.
Solving Logarithmic Equations
Once you've used the one-to-one property, you often have a simpler equation to work with. The next step involves solving the resulting algebraic equation. Let's go through the process:
Starting from our specific equation
  • \( 6 - m = 3m \)
The first step to solve for \( m \) is to get all terms containing \( m \) on one side of the equation.
  • Add \( m \) to both sides:
  • \( 6 = 4m \)
Now, divide both sides by 4 to isolate \( m \):
  • \( m = \frac{6}{4} \)
Finally, simplify the fraction to get:
  • \( m = \frac{3}{2} \)
This process of isolating the variable involves basic algebraic manipulation techniques which are key to solving equations.
Simplifying Fractions
When you solve an equation and end up with a fraction, simplifying it is the next step. Simplifying fractions makes them easier to interpret and use in further calculations. Here鈥檚 how to simplify the fraction from our example:
  • The fraction we obtained is \( \frac{6}{4} \).
  • This can be simplified by dividing both the numerator and denominator by their greatest common divisor (GCD), which in this case is 2.
The steps are as follows:
  • Divide 6 by 2 to get 3.
  • Divide 4 by 2 to get 2.
  • Thus, \( \frac{6}{4} \) simplifies to \( \frac{3}{2} \).
Simplifying fractions ensures that your answers are in their most straightforward and easily understandable form, which is important for both interpreting the result and using it in further mathematical operations.

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Most popular questions from this chapter

For the following exercises, refer to Table 4.31. $$\begin{array}{|c|c|c|c|c|c|c|c|c|}\hline x & {0} & {2} & {4} & {5} & {7} & {8} & {10} & {11} & {15} & {17} \\ \hline f(x) & {12} & {28.6} & {52.8} & {70.3} & {99.9} & {112.5} & {125.8} & {127.9} & {135.1} & {135.9} \\\ \hline\end{array}$$ To the nearest whole number, what is the predicted carrying capacity of the model?

Use logarithms to solve. \(3^{2 x+1}=7^{x-2}\)

For the following exercises, use this scenario: The population \(P\) of an endangered species habitat for wolves is modeled by the function \(P(x)=\frac{558}{1+54.8 e^{-0.462 x}},\) where \(x\) is given in years. Use the intersect feature to approximate the number of years it will take before the population of the habitat reaches half its carrying capacity.

For the following exercises, refer to Table 4.28. $$\begin{array}{|c|c|c|c|c|c|c|}\hline x & {1} & {2} & {3} & {4} & {5} & {6} \\\ \hline f(x) & {5.1} & {6.3} & {7.3} & {7.7} & {8.1} & {8.6} \\\ \hline\end{array}$$ Graph the logarithmic equation on the scatter diagram.

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