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Chapter 12: Problem 13

Solve. Where appropriate, include approximations to three decimal places. $$ 2^{x}=10 $$

Short Answer

Expert verified
x \approx 3.322

Step by step solution

01

- Take the logarithm of both sides

To solve the equation \( 2^x = 10 \), take the logarithm of both sides. You can use either common logarithm (log base 10) or natural logarithm (log base e). For this example, we will use the natural logarithm (ln). This gives us: \[ \ln(2^x) = \ln(10) \]
02

- Use the power rule for logarithms

Apply the power rule, which states \(\ln(a^b) = b \cdot \ln(a)\). Therefore: \[ x \cdot \ln(2) = \ln(10) \]
03

- Solve for x

To isolate x, divide both sides by \( \ln(2) \). This gives us: \[ x = \frac{\ln(10)}{\ln(2)} \]
04

- Calculate the value

Using a calculator, find the natural logarithms of 10 and 2, then divide them: \[ x = \frac{\ln(10)}{\ln(2)} \approx \frac{2.302585}{0.693147} \approx 3.322 \]

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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 special type of logarithm with the base \(e\). The number \(e\) is an irrational constant approximately equal to 2.71828. Natural logarithms are useful in many areas of mathematics, including solving exponential equations. When faced with an equation involving an exponent, like \(2^x = 10\), taking the natural logarithm of both sides can simplify the process.
For example: \(\text{ln}(2^x) = \text{ln}(10)\). This step is crucial because it allows us to utilize logarithmic properties to solve for the variable exponent.
power rule for logarithms
The power rule for logarithms is a handy tool when solving exponential equations. This rule states that \(\text{ln}(a^b) = b \text{ln}(a)\). It allows you to take an exponent out of the logarithm and turn it into a coefficient. For example, from the equation \(\text{ln}(2^x) = \text{ln}(10)\), we can apply the power rule to get: \(x \text{ln}(2) = \text{ln}(10)\). This simplifies the equation significantly and allows us to isolate the variable. Knowing the power rule is fundamental when dealing with logarithms and exponents in algebra and calculus.
solving exponential equations
Solving exponential equations involves a few logical steps to isolate the variable. Consider the equation \(2^x = 10\) and assume we need to solve for \x\. Here鈥檚 a simple roadmap:
  • Take the natural logarithm of both sides: \(\text{ln}(2^x) = \text{ln}(10)\).
  • Apply the power rule for logarithms: \(x \text{ln}(2) = \text{ln}(10)\).
  • Isolate the variable by dividing both sides by \(\text{ln}(2)\): \(x = \frac{\text{ln}(10)}{\text{ln}(2)}\).
  • Use a calculator to find the values: \(x \text{approx}= \frac{2.302585}{0.693147} \text{approx} 3.322\).
With this method, you can solve many types of exponential equations easily. Remember to always check your final answer with a calculator to ensure accuracy and understand each step鈥檚 significance.

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