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Chapter 1: Problem 257

For the following exercises, write the equation in equivalent logarithmic form. $$ 9^{0}=1 $$

Short Answer

Expert verified
\(\log_9(1) = 0\)

Step by step solution

01

Identify the Exponential Form

The given equation is in exponential form: \(9^0 = 1\). Here, the base is 9, the exponent is 0, and the result is 1.
02

Write the Logarithmic Form

To convert the exponential equation \(b^e = n\) to its equivalent logarithmic form, the equation is written as \(\log_b(n) = e\).
03

Apply the Conversion

Using the rule from Step 2, substitute the values from the given problem: base \(b = 9\), exponent \(e = 0\), and the result \(n = 1\). The logarithmic form is \(\log_9(1) = 0\).

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Key Concepts

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

Exponential Form
When we talk about the exponential form, we are referring to numbers expressed as a base raised to an exponent. It's a compact way to show repeated multiplication. For example, in the expression \(9^0\), "9" is the base and "0" is the exponent.
This means that the base, 9, is multiplied by itself zero times, which might sound a bit odd at first. However, remember that any number raised to the power of zero equals 1. Why? Because it's a fundamental property of exponents: no matter the base, any number with an exponent of zero is 1.
Exponential form:
  • Simplifies writing very large or very small numbers
  • Turns multiplication of the same number into an easier task
  • Is widely used in science and engineering to express large powers
Base and Exponent
The base and exponent are key elements in understanding mathematical expressions involving powers. Let's break them down:
The **base** is the number that is being multiplied. In our earlier example, 9 is the base. It tells us the number that is repeatedly being multiplied. The **exponent** tells us how many times to multiply the base by itself. In \(9^0\), the exponent is 0. This means 9 is multiplied zero times, resulting in a value of 1.
Understanding base and exponent is crucial because:
  • They form the foundation for other arithmetic operations like logarithms
  • Together, they explain exponential growth and decay found in real-world scenarios
  • Mastering them aids in learning more complex math concepts
Conversion of Equations
Converting an exponential equation into a logarithmic form is a common task in algebra. This is done to solve equations involving exponentials by finding unknown values of exponents.
To convert, recall the formula: if an exponential equation is written as \(b^e = n\), it turns into a logarithmic statement \(\log_b(n) = e\). In our example, \(9^0 = 1\) converts to \(\log_9(1) = 0\). Here, you simply transfer components: base remains base, result becomes the argument of the log, and the exponent goes to the other side of the equation.
The benefits of conversion are:
  • Logarithmic form provides an alternative perspective to solve complex equations
  • Simplifies calculations, especially when dealing with very large or small numbers
  • Examples include computing time in exponentially growing scenarios, like interest rates

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Most popular questions from this chapter

For the following exercises, sketch the graph of the logarithmic function. Determine the domain, range, and vertical asymptote. $$ f(x)=1-\ln x $$

For the following exercises, use the change-of-base formula and either base 10 or base \(e\) to evaluate the given expressions. Answer in exact form and in approximate form, rounding to four decimal places. $$ \log _{7} 82 $$

For the following problems, consider the population of Ocean City, New Jersey, which is cyclical by season. In reality, the overall population is most likely increasing or decreasing throughout each year. Let's reformulate the model as \(P(t)=82.5-67.5 \cos [(\pi / 6) t]+t, \quad\) where \(t\) is time in months \((t=0 \text { represents January } 1)\) and \(P\) is population (in thousands). When is the first time the population reaches \(200,000 ?\)

An investment is compounded monthly, , quarterly, or yearly and is given by the function \(A=P\left(1+\frac{j}{n}\right)^{n t}\) where \(A\) is the value of the investment at time \(t, P\) is the initial principle that was invested, \(j\) is the annual interest rate, and \(n\) is the number of time the interest is compounded per year. Given a yearly interest rate of 3.5\(\%\) and an initial principle of \(\$ 100,000\) , find the amount \(A\) accumulated in 5 years for interest that is compounded a. daily, b., monthly, c. quarterly, and d. yearly.

[T] A condominium in an upscale part of the city was purchased for \(\$ 432,000\) . In 35 years it is worth \(\$ 60,500\) . Find the rate of depreciation.
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