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Chapter 4: Problem 23

Write the equation in exponential form. $$ \log _{4} 1=0 $$

Short Answer

Expert verified
4^0 = 1

Step by step solution

01

Understand the Logarithmic Form

The given equation is in logarithmic form: \ \(\log_{4}{1} = 0\) \ This represents the exponent to which the base 4 must be raised to get 1.
02

Recall the Definition of Logarithms

By definition, \(\log_{b}{a} = c\) is equivalent to \(b^c = a\).
03

Convert to Exponential Form

Using the definition, convert the logarithmic equation \(\log_{4}{1} = 0\) to exponential form: \(4^0 = 1\).

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Key Concepts

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

logarithmic equations
Logarithmic equations are equations that involve logarithms. These can often look complex at first, but understanding their basic properties makes them much easier to handle. A logarithm basically answers the question: 'To what power must the base be raised, to produce a given number?' For instance, in \( \log_4{1} = 0 \), we are asking: 'To what power must 4 be raised to get 1?' This makes it a logarithmic equation.

When dealing with logarithmic equations, it's crucial to understand the properties and definitions of logarithms. Once we decode a logarithmic equation, it often becomes much simpler.
convert to exponential form
Converting a logarithmic equation to an exponential form can simplify our understanding of the problem. To recall, \( \log_b{a} = c \) is equivalent to \( b^c = a \). Let's break this down with our specific example:
  • We have the logarithmic equation: \( \log_{4}{1} = 0 \).
  • The base (b) is 4.
  • The exponent (c) is 0.
  • The result (a) is 1.
Using the conversion rule, we get \( 4^0 = 1 \). This tells us that raising 4 to the power of 0 results in 1, which makes sense, as any number raised to the power of 0 always equals 1.

Converting to the exponential form is very useful, especially in solving logarithmic equations, as it often makes the relationship between the numbers clearer and more intuitive.
properties of logarithms
To fully grasp logarithmic equations, we need to understand some key properties of logarithms:
  • Product Property: \( \log_b{(MN)} = \log_b{M} + \log_b{N} \). This property states that the logarithm of a product is the sum of the logarithms of the factors.
  • Quotient Property: \( \log_b{(M/N)} = \log_b{M} - \log_b{N} \). The logarithm of a quotient is the difference of the logarithms.
  • Power Property: \( \log_b{(M^p)} = p \log_b{M} \). The logarithm of a power is the exponent times the logarithm of the base.
  • Change of Base Formula: \( \log_b{M} = \frac{\log_c{M}}{\log_c{b}} \). This allows you to convert between different bases.
Knowing these properties can simplify many logarithmic problems. For example, in our initial problem \( \log_{4}{1} = 0 \), we suspect some trickery because logarithms are inherently complex. But using the properties and definitions, it transforms into something simple and elegant like \( 4^0 = 1 \).

So always keep these properties in mind as they are extremely handy tools!

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Most popular questions from this chapter

(See Example 8 ) a. Estimate the value of the logarithm between two consecutive integers. For example, \(\log _{2} 7\) is between 2 and 3 because \(2^{2}<7<2^{3}\). b. Use the change-of-base formula and a calculator to approximate the logarithm to 4 decimal places. c. Check the result by using the related exponential form. $$ \log _{2} 0.2 $$

Write \(10^{2 x-4}=80,600\) in logarithmic form.

9\. The population of the United States \(P(t)\) (in millions) since January 1,1900 , can be approximated by $$P(t)=\frac{725}{1+8.295 e^{-0.0165 t}}$$ where \(t\) is the number of years since January \(1,1900 .\) (See Example 6\()\) a. Evaluate \(P(0)\) and interpret its meaning in the context of this problem. b. Use the function to predict the U.S. population on January \(1,2020 .\) Round to the nearest million. c. Use the function to predict the U.S. population on January 1,2050 . d. Determine the year during which the U.S. population will reach 500 million. e. What value will the term \(\frac{8.295}{e^{0.0165 t}}\) approach as \(t \rightarrow \infty\) ? f. Determine the limiting value of \(P(t)\).

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. \(\log (p+17)=4.1\)

Write the domain in interval notation. $$ f(x)=\log _{5}(5-3 x) $$
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