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Chapter 6: Problem 48

Solve each exponential equation. Express irrational solutions in exact form. $$ 3^{x}=14 $$

Short Answer

Expert verified
x = \frac{\text{ln}(14)}{\text{ln}(3)}

Step by step solution

01

Take the Natural Logarithm of Both Sides

Start by taking the natural logarithm (ln) of both sides of the equation. This helps in bringing the exponent down using properties of logarithms.\[ \text{ln}(3^x) = \text{ln}(14) \]
02

Apply Logarithm Power Rule

Use the power rule of logarithms, which states that \( \text{ln}(a^b) = b \text{ln}(a) \). Apply this rule to bring the exponent \( x \) in front of the logarithm.\[ x \text{ln}(3) = \text{ln}(14) \]
03

Isolate the Variable \( x \)

To solve for \( x \), divide both sides of the equation by \( \text{ln}(3) \).\[ x = \frac{\text{ln}(14)}{\text{ln}(3)} \]
04

Simplify the Expression

This fraction represents the exact form of the solution since \( \text{ln}(14) \) and \( \text{ln}(3) \) are irrational numbers. \[ x = \frac{\text{ln}(14)}{\text{ln}(3)} \]

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Key Concepts

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

Natural Logarithm
The natural logarithm, denoted as \text{ln}, is a logarithm with the base of the mathematical constant \(e \approx 2.71828\). It plays a crucial role in higher mathematics and is particularly useful in solving exponential equations.
When solving equations involving exponential functions, we often use natural logarithms because they simplify the expression. For example, if you have an equation like \(3^x=14\), taking the natural logarithm of both sides can make it simpler to solve:
\text{ln}(3^x) = \text{ln}(14).
Here, log properties make it easier to handle the equation.
Logarithm Power Rule
The logarithm power rule states that \text{ln}(a^b) = b \text{ln}(a). This rule is very useful when dealing with exponents.
For example, consider the equation \text{ln}(3^x) = \text{ln}(14) from before. We apply the power rule to bring down the exponent \(x\):
\( x \text{ln}(3) = \text{ln}(14) \).
This way, the exponent \(x\) is no longer attached to the base \(3\) and is now multiplied by the logarithm, making it simpler to solve. This is a very common technique when solving exponential equations.
Isolating the Variable
Once you have applied the logarithm power rule, you need to isolate the variable to solve for it. For instance, in the equation \text{ln}(3^x) = \text{ln}(14), after applying the power rule, we get \(x \text{ln}(3) = \text{ln}(14)\).
To isolate \(x\), divide both sides by \text{ln}(3):
\( x = \[ \frac{\text{ln}(14)}{\text{ln}(3)} \] \).
Isolating the variable is essential in solving equations because it leaves you with the solution \(x\) on one side. In this example, the fraction \frac{\text{ln}(14)}{\text{ln}(3)} represents the exact solution for \(x\), as both \text{ln}(14) and \text{ln}(3) are considered to be irrational.

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Most popular questions from this chapter

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