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Chapter 5: Problem 32

Use the definition of the logarithmic function to find \(x\). $$ \begin{array}{ll}{\text { (a) } \log _{4} 2=x} & {\text { (b) } \log _{4} x=2}\end{array} $$

Short Answer

Expert verified
(a) x = 1/2; (b) x = 16.

Step by step solution

01

Understand the Definition of Logarithm

The logarithm \( \log_b a = c \) is defined such that \( b^c = a \). It states that \( b \) raised to the power of \( c \) equals \( a \). We'll apply this to solve for \( x \).
02

Solve Part (a) \( \log_4 2 = x \)

Using the logarithm definition, set \( 4^x = 2 \). This equation means the base 4 raised to the power of \( x \) will equal 2. To solve for \( x \), we need to find a power of 4 that results in 2. Recognizing 2 as an adjustment of \( 4^{1/2} \) (since \( 2 = \sqrt{4} \)), we find \( x = \frac{1}{2} \).
03

Solve Part (b) \( \log_4 x = 2 \)

Set the equation according to the definition: \( 4^2 = x \). Calculate \( 4^2 \) which equals 16, thus \( x = 16 \).

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Key Concepts

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

Definition of Logarithms
Understanding the definition of logarithms is crucial for solving equations that involve them. A logarithm answers the question: "To what power must the base be raised, to produce a given number?" For instance, in the expression \( \log_b a = c \),
  • \( b \) is the base of the logarithm
  • \( a \) is the number you want to find the logarithm for
  • \( c \) is the exponent or power that the base must be raised to, in order to equal \( a \)
So, log\( _b \) \( a \) = \( c \) means \( b^c = a \). This is the fundamental relationship that defines logarithms. It's a powerful tool that lets us transform multiplicative situations into additive ones, simplifying complex mathematical problems. Think of it as the inverse operation of exponentiation.
Solving Logarithmic Equations
Solving logarithmic equations involves applying the definition of logarithms to rewrite the log equation as an exponential equation. For example, if you have \( \log_4 2 = x \), you rewrite it using the fact that it means \( 4^x = 2 \).
  • You determine that \( 4^x \) should equal 2.
  • Knowing that \( 2 = 4^{1/2} \) since 2 is the square root of 4, helps you see that \( x = \frac{1}{2} \).
For another case, such as \( \log_4 x = 2 \), the definition yields \( 4^2 = x \), leading directly to the fact that \( x = 16 \). The procedure effectively finds 鈥榎( x \)鈥� by determining what power (in the first case) or result (in the second case) aligns with the logarithmic definition when applied to a specific base.
Properties of Exponents
Exponents are closely linked with logarithms and share fundamental properties that are very useful when handling logarithmic equations.
  • Multiplication: \( a^m \times a^n = a^{m+n} \)
  • Division: \( \frac{a^m}{a^n} = a^{m-n} \)
  • Power to a Power: \((a^m)^n = a^{m \times n} \)
  • Root: \( a^{1/n} = \sqrt[n]{a} \)
These properties help you manipulate expressions when solving logarithmic equations. For example, knowing how to express roots as fractional exponents allows you to identify that \( 4^{1/2} = 2 \), which can simplify finding solutions to equations involving logs. Recognizing these connections is key to understanding and working effectively with exponential and logarithmic expressions.

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Most popular questions from this chapter

Bacteria Culture A culture starts with 8600 bacteria. After one hour the count is \(10,000\) . (a) Find a function that models the number of bacteria \(n(t)\) after \(t\) hours. (b) Find the number of bacteria after 2 hours. (c) After how many hours will the number of bacteria double?

Nancy wants to invest \(\$ 4000\) in saving certificates that bear an interest rate of 9.75\(\%\) per year, compounded semiannully. How long a time period should she choose to save an amount of \(\$ 5000 ?\)

Carbon Dating The age of an ancient artifact can be determined by the amount of radioactive carbon-14 remaining in it. If \(D_{0}\) is the original amount of carbon- 14 and \(D\) is the amount remaining, then the artifact's age \(A\) (in years) is given by $$ A=-8267 \ln \left(\frac{D}{D_{0}}\right) $$ Find the age of an object if the amount \(D\) of carbon \(-14\) that remains in the object is 73\(\%\) of the original amount \(D_{0}\) .

\(17-24\) . These exercises use the radioactive decay model. Radioactive Radon After 3 days a sample of radon- 222 has decayed to 58\(\%\) of its original amount. (a) What is the half-life of radon- 222\(?\) (b) How long will it take the sample to decay to 20\(\%\) of its original amount?

Suppose you鈥檙e driving your car on a cold winter day \(\left(20^{\circ} \mathrm{F} \text { outside) and the engine overheats (at }\right.\) about \(220^{\circ} \mathrm{F}\) ). When you park, the engine begins to cool down. The temperature \(T\) of the engine \(t\) minutes after you park satisfies the equation $$\ln \left(\frac{T-20}{200}\right)=-0.11 t$$ (a) Solve the equation for \(T\) . (b) Use part (a) to find the temperature of the engine after \(20 \min (t=20) .\)
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