Question

Use a calculator to evaluate each expression. Round your answer to three decimal places. $$ \frac{\ln 4+\ln 2}{\log 4+\log 2} $$

Short answer

2.302

Step-by-step solution

Apply the properties of logarithms

Use the properties of logarithms to simplify the expression. Recall that \(\text{ln}(a) + \text{ln}(b) = \text{ln}(ab) \) and \( \text{log}(a) + \text{log}(b) = \text{log}(ab) \). Hence, \(\frac{\text{ln} 4 + \text{ln} 2}{\text{log} 4 + \text{log} 2} = \frac{\text{ln}(4 \times 2)}{\text{log}(4 \times 2)}= \frac{\text{ln} 8}{\text{log} 8}\).

Evaluate natural and common logarithms

Use a calculator to evaluate \( \text{ln} 8 \) and \( \text{log} 8 \). \( \text{ln} 8 \approx 2.079 \) and \( \text{log} 8 \approx 0.903 \).

Divide the logarithms

Calculate the value by dividing the two logarithms: \(\frac{2.079}{0.903} \approx 2.302\).

Key concepts

natural logarithms

Natural logarithms are logarithms with the base of Euler's number, e (approximately 2.71828). They are expressed as \(\text{ln}(x)\). Natural logarithms are commonly used in fields like biology, economics, and complex calculations due to their natural growth properties.
For example, to find the natural logarithm of 8 (\(\text{ln} 8\)), you can use a calculator. You'll find that \(\text{ln} 8 \approx 2.079\).

Natural logarithms can simplify many mathematical expressions and are interconnected with exponential functions. They also help in solving equations involving growth rates and decay processes.

common logarithms

Common logarithms are logarithms with base 10, denoted as \(\text{log}(x)\). These are widely used in various scientific and engineering disciplines. The base 10 logarithms are handy for dealing with large numbers since our numerical system is decimal.
For instance, calculating the common logarithm of 8 (\(\text{log} 8\)) yields approximately 0.903 using a calculator.

Common logarithms simplify complex multiplications into additions, which are easier to perform. They also play a crucial role in understanding phenomena in acoustics, earthquake magnitude scales, and more.

logarithm properties

Understanding the properties of logarithms is essential. They allow us to simplify and manipulate logarithmic expressions effectively. Some important properties include:

• Sum Property: \(\text{ln}(a) + \text{ln}(b) = \text{ln}(ab)\) and \(\text{log}(a) + \text{log}(b) = \text{log}(ab)\).
  This property helps to combine logarithms of multiplication into single expressions.
• Quotient Property: \(\text{ln}(a) - \text{ln}(b) = \text{ln}\frac{a}{b}\) and \(\text{log}(a) - \text{log}(b) = \text{log}\frac{a}{b}\).
• Power Property: \(\text{ln}(a^b) = b\text{ln}(a)\) and \(\text{log}(a^b) = b\text{log}(a)\).

Using these properties simplifies our original expression to \( \frac{\text{ln} 8}{\text{log} 8} \), making it easier to calculate.

calculator usage

To solve logarithmic expressions, a calculator can be an invaluable tool. Most scientific calculators have dedicated buttons for natural logarithms (\(\text{ln}\)) and common logarithms (\(\text{log}\)).
Here's how to use a calculator efficiently:

• Input the number and press the \(\text{ln}\) button to get the natural logarithm.
• Input the number and press the \(\text{log}\) button to get the common logarithm.

In our exercise, you would:

• Input 8, press \(\text{ln}\), which gives approximately 2.079.
• Input 8, press \(\text{log}\), which results in approximately 0.903.
• Divide these results: \( \frac{2.079}{0.903} \approx 2.302\).

Getting comfortable with these steps ensures you can tackle complex expressions confidently.