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Chapter 3: Problem 44

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$\ln x+\ln 3$$

Short Answer

Expert verified
The condensed form of the expression is \( \ln(3x) \).

Step by step solution

01

Identify the Logarithmic Property

The logarithmic property that applies in this scenario states that - if we have two logs of the same base separated by a plus sign: \( \log_b(m) + \log_b(n) \), this is the same as a single log whose argument is the product of the original arguments: \( \log_b(m*n) \). This exercise has provided two natural logarithms (which have the base 'e') - \( \ln(x) \) and \( \ln(3) \), therefore, it can be simplified into a single natural log expression.
02

Apply the Logarithmic Property

Apply the identified property to the given expression: \( \ln(x) + \ln(3) \) to get a single logarithmic expression. This results in \( \ln(x*3) \) or simply \( \ln(3x) \).

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Key Concepts

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

Understanding Properties of Logarithms
Logarithms possess fascinating and useful properties that allow us to manipulate and simplify expressions. These properties can make calculations easier and help represent complex expressions more straightforwardly.
  • Product Property: One of the most useful properties is the product property, which states: when you add two logarithms with the same base, you can combine them into a single logarithm. Specifically, \[\log_b(m) + \log_b(n) = \log_b(m \times n)\]

  • Quotient Property: If you're subtracting two logarithms of the same base, you can use the quotient rule: \[\log_b(m) - \log_b(n) = \log_b\left(\frac{m}{n}\right)\]

  • Power Property: Lastly, the power rule tells us: to bring a coefficient in front of a log inside, raising the log's argument to that power, like so: \[c\cdot \log_b(m) = \log_b(m^c)\]
Understanding these properties allows for manipulating logs into simpler forms, which are especially handy when solving equations or integrating functions.
Natural Logarithms Explained
Natural logarithms are a special kind of logarithm that use 'e' (approximately 2.718) as their base. They are represented as \( \ln(x) \), which is equivalent to \( \log_e(x) \). Why 鈥榚鈥�? Because 'e' is a fundamental constant in mathematics, much like \( \pi \).
  • It is crucial in the continuous growth processes, and natural logs arise in the compound interest, population growth models, and more.

  • Natural logs come in handy for solving calculus problems involving derivatives and integrals, as they simplify handling exponential functions.
Using natural logarithms, mathematicians can easily depict phenomena of exponential change, giving a clearer understanding when interpreting exponential growth or decay processes.
Condensing Logarithmic Expressions
The art of condensing logarithmic expressions revolves around simplifying complex log equations into a single, manageable form. This is essential when working with lengthy or intricate logarithmic expressions.
  • The key is to apply the right properties of logarithms, like the product, quotient, and power properties we talked about.

  • Consider the example: \( \ln(x) + \ln(3) \). You can combine these using the product rule to get \( \ln(3x) \).

  • This process aids in solving logarithmic equations or evaluating them without a calculator whenever possible.
Condensing makes calculation processes less cumbersome and is instrumental in various fields involving mathematical models or analyses.

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Most popular questions from this chapter

In parts (a)-(c), graph \(f\) and \(g\) in the same viewing rectangle. a. \(f(x)=\ln (3 x), g(x)=\ln 3+\ln x\) b. \(f(x)=\log \left(5 x^{2}\right), g(x)=\log 5+\log x^{2}\) c. \(f(x)=\ln \left(2 x^{3}\right), g(x)=\ln 2+\ln x^{3}\) d. Describe what you observe in parts (a)-(c). Generalize this observation by writing an equivalent expression for \(\log _{b}(M N),\) where \(M>0\) and \(N>0\) e. Complete this statement: The logarithm of a product is equal to _______________.

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$\log (x+3)+\log x=1$$

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$5^{x}=3 x+4$$

The \(p H\) scale is used to measure the acidity or alkalinity of a solution. The scale ranges from 0 to \(14 .\) A neutral solution, such as pure water, has a pH of 7. An acid solution has a pH less than 7 and an alkaline solution has a pH greater than 7. The lower the \(p H\) below 7 , the more acidic is the solution. Each whole-number decrease in \(p H\) represents a tenfold increase in acidity. (GRAPH CAN'T COPY). The \(p H\) of a solution is given by $$\mathrm{pH}=-\log x$$ where \(x\) represents the concentration of the hydrogen ions in the solution, in moles per liter. Use the formula to solve. Express answers as powers of \(10 .\) a. Normal, unpolluted rain has a pH of about 5.6. What is the hydrogen ion concentration? b. An environmental concern involves the destructive effects of acid rain. The most acidic rainfall ever had a \(\mathrm{pH}\) of \(2.4 .\) What was the hydrogen ion concentration? c. How many times greater is the hydrogen ion concentration of the acidic rainfall in part (b) than the normal rainfall in part (a)?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because logarithms are exponents, the product, quotient, and power rules remind me of properties for operations with exponents.
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