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Chapter 10: Problem 32

Use a calculator to approximate each logarithm to four decimal places. $$\log _{2} 15$$

Short Answer

Expert verified
\(\text{log}_{2} 15 \approx 3.9069\)

Step by step solution

01

Understanding the Logarithm

The exercise asks to approximate \(\text{log}_{2} 15\) to four decimal places. This means finding the exponent to which 2 must be raised to obtain 15.
02

Change of Base Formula

Use the change of base formula for logarithms: \(\text{log}_{b} a = \frac{\text{log} a}{\text{log} b}\). This allows the logarithm to be converted to base 10, which can be used with a calculator.
03

Apply the Formula

Apply the change of base formula with \(\text{log}_{2} 15\): \(\text{log}_{2} 15 = \frac{\text{log} 15}{\text{log} 2}\).
04

Use a Calculator

Use the calculator to find \(\text{log} 15\) and \(\text{log} 2\): \(\text{log} 15 \approx 1.1761\) and \(\text{log} 2 \approx 0.3010\).
05

Calculate the Result

Divide the results from Step 4: \(\frac{1.1761}{0.3010} \approx 3.9069\).
06

Round to Four Decimal Places

Ensure the result is rounded to four decimal places. The computed value \(\text{log}_{2} 15 \approx 3.9069\) is already to four decimal places.

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Key Concepts

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

Change of Base Formula
The change of base formula is a very useful tool in mathematics. It allows us to convert logarithms from one base to another, making it easier to use a calculator for the calculations. The formula is defined as: \[ \log_b a = \frac{\log a}{\log b} \]Here, \( \log b \) represents the logarithm of the base to which we want to convert. For example, if you want to calculate \( \log_2 15 \), you can convert it into a base-10 logarithm (which is commonly available on calculators) using the change of base formula: \[ \log_2 15 = \frac{\log 15}{\log 2} \] By transforming the base, the computation of non-standard logarithms becomes straightforward and convenient.
Logarithmic Functions
Logarithmic functions are the inverse of exponential functions. In simpler terms, if you have an equation \( y = b^x \), then its logarithmic form would be \( x = \log_b y \). The logarithm \( \log_b y \) answers the question: **To what power must the base \( b \) be raised to produce the number \( y \)?** Logarithms have several properties that make them extremely useful in mathematics:
  • The product rule: \( \log_b (MN) = \log_b M + \log_b N \)
  • The quotient rule: \( \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \)
  • The power rule: \( \log_b (M^p) = p \cdot \log_b M \)
Understanding these properties helps in simplifying logarithmic equations and solving real-world problems efficiently.
Using a Calculator for Logarithms
Calculators are handy tools for quickly computing logarithms, especially those with base-10 (common logarithms) and base-e (natural logarithms). When using a calculator to find logarithms, simply follow these steps:
  1. Identify the logarithm you need to compute. For example, \( \log_2 15 \).
  2. Convert the original base to base-10 using the change of base formula: \( \log_2 15 = \frac{\log 15}{\log 2} \).
  3. Use the calculator to find \( \log 15 \) and \( \log 2 \). Once you input these values, you can get the approximate values: \( \log 15 \approx 1.1761 \) and \( \log 2 \approx 0.3010 \).
  4. Divide the results from the calculator: \( \frac{1.1761}{0.3010} \approx 3.9069 \).
  5. Round the final result to the required number of decimal places. In this case, 4 decimal places: \( 3.9069 \).
Using these steps, you can easily approximate logarithms with the help of a calculator, ensuring accurate and efficient computations.

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Most popular questions from this chapter

Solve each equation. Give exact solutions. $$ \log _{5}(3 t+2)-\log _{5} t=\log _{5} 4 $$

To four decimal places, the values of \(\log _{10} 2\) and \(\log _{10} 9\) are $$\log _{10} 2=0.3010 \text { and } \log _{10} 9=0.9542$$ Use these values and the properties of logarithms to evaluate each expression. DO NOT USE A CALCULATOR. See Example 5. $$ \log _{10} 2^{19} $$

Find the amount of money in an account after 8 yr if \(\$ 4500\) is deposited at \(6 \%\) annual interest compounded as follows. (a) Annually (b) Semiannually (c) Quarterly (d) Daily (Use \(n=365 .)\) (e) Continuously

To four decimal places, the values of \(\log _{10} 2\) and \(\log _{10} 9\) are $$\log _{10} 2=0.3010 \text { and } \log _{10} 9=0.9542$$ Use these values and the properties of logarithms to evaluate each expression. DO NOT USE A CALCULATOR. See Example 5. $$ \log _{10} 162 $$

Determine whether common logarithms or natural logarithms would be a better choice to use for solving each equation. Do not actually solve. $$ 10^{0.0025 x}=75 $$
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