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Chapter 12: Problem 10

Solve each equation. Approximate solutions to three decimal places. $$ 5^{3 x}=11 $$

Short Answer

Expert verified
x 鈮� 0.497

Step by step solution

01

Take the natural logarithm of both sides

To solve the equation, start by taking the natural logarithm (ln) of both sides: \( \ln(5^{3x}) = \ln(11) \).
02

Use logarithmic properties

Apply the logarithmic property \( \ln(a^b) = b\ln(a) \) to rewrite the left side: \( 3x \ln(5) = \ln(11) \).
03

Isolate the variable x

Divide both sides by \( 3 \ln(5) \) to isolate \( x \): \( x = \frac{\ln(11)}{3 \ln(5)} \).
04

Calculate using a calculator

Use a calculator to find the approximate values of the natural logarithms: \( \ln(11) \approx 2.398 \) and \( \ln(5) \approx 1.609 \). After calculation: \( x \approx \frac{2.398}{3 \times 1.609} \approx \frac{2.398}{4.827} \approx 0.497 \).

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Key Concepts

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

Natural Logarithm
A natural logarithm, denoted as \( \text{ln} \), is a logarithm with the base of Euler's number \( e \), where \( e \approx 2.71828 \). The natural logarithm is useful because it simplifies the process of dealing with exponential equations.
This makes equations like \( a^b \) easier to manage.

For example, in the equation \( 5^{3x} = 11 \), taking the natural logarithm of both sides gives you:
\( \text{ln}(5^{3x}) = \text{ln}(11) \).

This step helps us convert the exponential term into a form that can be simplified using properties of logarithms.
Logarithmic Properties
Logarithmic properties are rules that apply to logarithms, making it easier to solve equations. One key property is \( \text{ln}(a^b) = b\text{ln}(a) \).
This property helps break down powers inside the logarithm.
In our exercise, we apply this property to \( \text{ln}(5^{3x}) \), transforming it into:
\( 3x \text{ln}(5) \).

This step is crucial as it allows us to move the exponent (3x) outside the log, simplifying the equation. We then solve for x:
  • \(3x \text{ln}(5) = \text{ln}(11)\)
  • \(x = \frac{\text{ln}(11)}{3 \text{ln}(5)}\)
This illustrates how understanding properties of logarithms can make solving such equations much more approachable.
Calculator Usage
Using a scientific calculator is essential to find approximate values of natural logarithms. Here鈥檚 how to do it:
  • Input the number you want the natural logarithm of. For \( \text{ln}(11) \), enter 11.
  • Press the \text{ln} button to get the value, which is approximately 2.398.
  • Repeat for \( \text{ln}(5) \), yielding approximately 1.609.
Now, you can use these values to solve for x:
\( x \frac{2.398}{3 \times 1.609} \ x \frac{2.398}{4.827} \ x \ 0.497.\)
With practice, this process becomes straightforward, making it easier to handle logarithmic and exponential equations independently.

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Most popular questions from this chapter

Each of the following functions is one-to-one. Graph the function as a solid line (or curve), and then graph its inverse on the same set of axes as a dashed line (or curve). Complete any tables to help graph the functions. $$ f(x)=-4 x $$

Solve each equation. \(\log _{10} x=-2\)

Use the special properties of logarithms to evaluate each expression. \(\log _{4} 4^{9}\)

Use the special properties of logarithms to evaluate each expression. \(\log _{2} 2^{-1}\)

Solve each equation. Approximate solutions to three decimal places. $$ 3^{2 x+1}=5^{x-1} $$
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