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Chapter 0: Problem 47

Rewrite the expression as a single logarithm. $$\frac{1}{2} \ln 4-\ln 2$$

Short Answer

Expert verified
\(\ln 1\)

Step by step solution

01

Applying the Power Rule

The power rule of logarithms states that \(a \log_b m = \log_b (m^a)\). Therefore, the given expression \(\frac{1}{2} \ln 4 - \ln 2\) can be rewritten as \(\ln 4^{1/2} - \ln 2\). This simplifies to \( \ln 2 - \ln 2\).
02

Applying the Division Rule

The division rule of logarithms states that \(\log_b m - \log_b n = \log_b (m/n)\). Therefore, the expression \(\ln 2 - \ln 2\) can be rewritten as \(\ln (2/2)\). This simplifies to \(\ln 1\).

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Key Concepts

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

Power Rule of Logarithms
The power rule of logarithms is a handy tool for manipulating logarithmic expressions. It tells us that if you have the logarithm of a base, raised to a power, you can move that exponent to multiply the logarithm. More simply, for any logarithm \(a \log_b(m)\) can be expressed as \(\log_b(m^a)\).
Imagine juggling with smaller numbers. If you have an expression like \(\frac{1}{2} \ln 4\), ride the power rule wave. Rewrite it to \(\ln 4^{1/2}\), or more familiarly, as \(\ln \sqrt{4}\). Since the square root of 4 is 2, it simplifies to \(\ln 2\).
In this step, understand the magic of sliding the exponent:
  • Your expression becomes cleaner.
  • It transforms multiplication into a more digestible form.
Breaking down exponents makes solving logarithms a breeze.
Division Rule of Logarithms
The division rule of logarithms is a clever trick to simplify differences of logarithms. Whenever you encounter an expression where one logarithm is subtracted from another, say \(\log_b m - \log_b n\), you can condense it into a single log by dividing the numbers inside it: \(\log_b \left(\frac{m}{n}\right)\).
Consider the expression \(\ln 2 - \ln 2\). Applying the division rule, it becomes \(\ln \left(\frac{2}{2}\right)\). Since \(2/2\) equals 1, you're simply left with \(\ln 1\).
Some key points to remember about this rule:
  • This process simplifies logarithmic expressions to involve fewer terms.
  • Division rule鈥檚 power is its ability to unify the logs into one concise form.
This step makes subtraction of logs feel like a shortcut to simplicity.
Simplifying Logarithmic Expressions
Simplifying logarithmic expressions involves using rules we know about logarithms to reduce them to their simplest form. The goal with simplification is to achieve a single, manageable expression.
In our example, the expression began as \(\frac{1}{2} \ln 4 - \ln 2\). Using the power rule and the division rule, we systematically simplified it step-by-step to \(\ln 1\).
Why is organizing and simplifying such expressions important?
  • Simplification helps make calculations more intuitive.
  • It's easier to interpret and use in further mathematical operations.
When you see complex expressions, envision how these rules can unfold to bring it back to basics, just like stripping a heavy coat to find clarity. Remember, every rule in logarithms exists to make things simpler and clearer!

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Most popular questions from this chapter

In golf, the goal is to hit a ball into a hole of diameter 4.5 inches. Suppose a golfer stands \(x\) feet from the hole trying to putt the ball into the hole. A first approximation of the margin of error in a putt is to measure the angle \(A\) formed by the ray from the ball to the right edge of the hole and the ray from the ball to the left edge of the hole. Find \(A\) as a function of \(x .\)

Adjust the graphing window to identify all vertical asympotes. $$f(x)=\frac{3}{x-1}$$

The Richter magnitude \(M\) of an earthquake is defined in terms of the energy \(E\) in joules released by the earthquake, with \(\log _{10} E=4.4+1.5 M .\) Find the energy for earthquakes with magnitudes (a) \(4,\) (b) 5 and (c) \(6 .\) For each increase in \(M\) of 1 by what factor does \(E\) change?

Use a triangle to simplify each expression. Where applicable, state the range of \(x\) 's for which the simplification holds. $$\cot \left(\cos ^{-1} x\right)$$

Find all vertical asymptotes. $$f(x)=\frac{4 x}{x^{2}+3 x-10}$$
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