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Chapter 3: Problem 68

Use the definition of a logarithm to write the exponential equation \(a^{x+2}=b\) in logarithmic form.

Short Answer

Expert verified
The exponential equation \(a^{x+2}=b\) can be written in logarithmic form as \(\log_a b = x+2\).

Step by step solution

01

Understanding Logarithmic and Exponential Form

Recall the relationship between logarithmic and exponential form. If we have an equation in exponential form \(a^c = b\), we can rewrite it in logarithmic form as \(\log_a b = c\). The base \(a\) and the value \(b\) remain the same in both forms, whereas the exponent \(c\) becomes the result in the logarithmic form.
02

Converting the Given Equation to Logarithmic Form

For the given equation \(a^{x+2}=b\), apply the conversion formula. Thus, the expression is rewritten as \(\log_a b = x+2\). In this logarithmic form, the base \(a\) in the original equation remains the base in the logarithm, \(b\) is the value we are taking the logarithm of, and \(x+2\) is the result of the logarithm.

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Key Concepts

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

Understanding Logarithms
Logarithms are the inverse operation of exponentiation, which means they undo what exponents do. An easy way to understand this is by thinking of logarithms as 'exponent detectives'鈥攖hey help you find the missing exponent you would need to get a certain number when using a specific base.

Let's consider the base 10 logarithm, which is common and often written as \(\log\). Saying \(\log(100)\) is 2 implies that 10 raised to the power of 2 equals 100. This can be expressed as \(10^2 = 100\). In general terms, if \(\log_b(x) = y\), this tells us that \(b^y = x\). Here, \(b\) is the base, \(x\) is the number you're taking the logarithm of, and \(y\) is the power to which the base must be raised to get \(x\).
Exponential Equations
Exponential equations are equations in which variables appear as exponents and require a particular base to be raised to a power, resulting in a specific value. These equations are of the form \(a^x = b\), where \(a\) is a positive number known as the base, \(x\) is the exponent or power, and \(b\) is the resulting number.

For instance, when we see \(2^3 = 8\), it is an exponential equation where 2 is the base raised to the power of 3, yielding the result 8. Solving these types of equations often involves finding the value of the exponent. In many cases, especially when the numbers do not work out to be whole numbers, logarithms come to the rescue to help us find the value of \(x\).
Logarithm Conversion
Converting between exponential and logarithmic forms is a fundamental skill, as it allows us to solve equations that would be challenging or impossible to solve in their original form. This conversion utilizes the definition of a logarithm.

Starting with an exponential equation like \(a^x = b\), you can convert to logarithmic form by applying the principle that \(\log_a(b) = x\). The base \(a\) of the exponent becomes the base of the log, the number \(b\) is what you input into the log function, and \(x\) is the log's output.

For example, the exponential equation \(3^4 = 81\) can be written in logarithmic form as \(\log_3(81) = 4\). This is saying that the power you need to raise 3 to get 81 is 4. These conversions are not just mathematical tricks鈥攖hey're essential tools for solving a wide variety of problems in fields like science, engineering, and finance.

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Most popular questions from this chapter

Solve \(0.85=0.5^{t / 5730}\) for \(t .\) Round to the nearest ten. [3.5]

Explain how to use the graph of the first function \(f\) to produce the graph of the second function \(F\). $$f(x)=\left(\frac{1}{3}\right)^{x}, F(x)=2\left[\left(\frac{1}{3}\right)^{x}\right]$$

Starting on the left side of a standard 88 -key piano, the frequency, in vibrations per second, of the \(n\) th note is given by \(f(n)=(27.5) 2^{(n-1) / 12}\) a. Using this formula, determine the frequency, to the nearest hundredth of a vibration per second, of middle C, key number 40 on an 88 -key piano. b. Is the difference in frequency between middle C (key number 40 ) and \(D\) (key number 42 ) the same as the difference in frequency between \(\mathrm{D}\) (key number 42 ) and \(\mathrm{E}\) (key number 44 )? Explain.

Involve the factorial function \(x !\), which is defined for whole numbers \(x\) as $$ x !=\left\\{\begin{array}{ll} 1, & \text { if } x=0 \\ x \cdot(x-1) \cdot(x-2) \cdot \cdots \cdot \cdot 3 \cdot 2 \cdot 1, & \text { if } x \geq 1 \end{array}\right. $$ For example, \(3 !=3 \cdot 2 \cdot 1=6\) and \(5 !=5 \cdot 4 \cdot 3 \cdot 2 \cdot 1=120\) During the 30 -minute period before a Broadway play begins, the members of the audience arrive at the theater at the average rate of 12 people per minute. The probability that \(x\) people will arrive during a particular minute is given by \(P(x)=\frac{12^{x} e^{-12}}{x !} .\) Find the probability, to the nearest \(0.1 \%\) that a. 9 people will arrive during a given minute. b. 18 people will arrive during a given minute.

If we assume that air resistance is proportional to the square of the velocity, then the velocity \(v\) in feet per second of an object \(t\) seconds after it has been dropped is given by $$v=50\left(\frac{e^{1.6 t}-1}{e^{1.6 t}+1}\right)$$ a. In how many seconds will the velocity be 20 feet per second? b. Determine the horizontal asymptote for the graph of this function. c. Write a sentence that describes the meaning of the horizontal asymptote in the context of this problem.
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