Question

Use the Change-of-Base Formula and a calculator to evaluate each logarithm. Round your answer to three decimal places. \(\log _{\pi} \sqrt{2}\)

Short answer

0.303

Step-by-step solution

Recall the Change-of-Base Formula

The Change-of-Base Formula allows you to evaluate logarithms with different bases using common logs (base 10) or natural logs (base e). The formula is: \[ \text{log}_b(a) = \frac{\text{log}(a)}{\text{log}(b)} \] Use this formula to change the base of the given logarithm to a base of your choice, primarily base 10 or base e.

Apply the Formula

Apply the Change-of-Base Formula to \[ \text{log}_{\text{π}}(\text{√2}) \] Substituting into the formula, you get: \[ \text{log}_{\text{π}}(\text{√2}) = \frac{\text{log}(\text{√2})}{\text{log}(\text{π})} \]

Simplify the Numerator

Recall that \[ \text{√2} = 2^{1/2} \] Using properties of logarithms: \[ \text{log}(2^{1/2}) = \frac{1}{2} \text{log}(2) \]

Substitute and Evaluate

Substitute \[ \text{log}(\text{√2}) \] back into the formula: \[ \frac{\frac{1}{2} \text{log}(2)}{\text{log}(\text{π})} \] Simplify to get: \[ \frac{1}{2} \frac{\text{log}(2)}{\text{log}(\text{π})} \]

Calculate the Logs Using a Calculator

Using a calculator, find the common logarithms: \[ \text{log}(2) \approx 0.301 \ \text{log}(\text{π}) \approx 0.497 \]

Compute the Final Value

Substitute these values into the equation and compute: \[ \frac{1}{2} \frac{0.301}{0.497} \approx \frac{1}{2} \times 0.606 \approx 0.303 \]

Key concepts

logarithmic functions

Logarithmic functions are the inverse of exponential functions. If we have an exponential function written as \( y = b^x \), where \( b \) is the base and \( x \) is the exponent, the corresponding logarithmic function is \(\text{log}_b(y) = x\). This means that the logarithm of \( y \) with base \( b \) gives us the exponent \( x \). Logarithmic functions are widely used in various fields such as mathematics, science, and engineering because they allow us to solve equations where the unknown value is an exponent.

properties of logarithms

Understanding the properties of logarithms is essential for simplifying and solving logarithmic equations. Here are some key properties:

• Product Rule: \( \text{log}_b(M \times N) = \text{log}_b(M) + \text{log}_b(N) \) - The logarithm of a product is the sum of the logarithms.
• Quotient Rule: \( \text{log}_b(\frac{M}{N}) = \text{log}_b(M) - \text{log}_b(N) \) - The logarithm of a quotient is the difference of the logarithms.
• Power Rule: \( \text{log}_b(M^k) = k \times \text{log}_b(M) \) - The logarithm of a power is the exponent times the logarithm.

These properties help to break down complex logarithmic expressions into simpler parts, making calculations and transformations more manageable.

calculator usage in logarithms

When evaluating logarithms using a calculator, it's important to be familiar with its logarithmic functions. Most calculators have buttons for both common logarithms (base 10, labeled as 'log') and natural logarithms (base e, labeled as 'ln'). For example, to find \( \text{log}(2) \) on your calculator:

• Press the 'log' button.
• Enter the number 2.
• Press the equals or enter button to get the result (≈ 0.301).

Similarly, you use the 'ln' button for natural logarithms. In problems requiring other bases, such as \( \text{log}_{\text{π}}(\text{√2}) \), you use the Change-of-Base Formula to convert to common logs or natural logs first.

base conversion in logarithms

The Change-of-Base Formula is an important tool for converting logarithms from one base to another. The formula is given by: \( \text{log}_b(a) = \frac{\text{log}(a)}{\text{log}(b)} \), where we typically use base 10 or base e for the logs in the numerator and denominator. For example, to evaluate \( \text{log}_{\text{π}}(\text{√2}) \), we convert it using the formula: \( \frac{\text{log}(\text{√2})}{\text{log}(\text{π})} \). By finding these common logs with a calculator and substituting the values, we can then compute the result. This conversion simplifies the process by allowing us to use a consistent base, often more familiar or accessible for computation.