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Chapter 3: Problem 6

$$\text { Explain why } b^{x}=e^{x \ln b}$$

Short Answer

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Question: Explain why \(b^x = e^{x \ln b}\) and its relationship with the exponential and natural logarithm functions. Answer: The equation \(b^x = e^{x \ln b}\) demonstrates the connection between the exponential function (b raised to the power of x) and the natural logarithm function (with base e). This relationship is derived using the properties of logarithms and exponential functions. The equation, \(b^x = e^{x \ln b}\), states that raising the base b to the power x is equivalent to raising e to the power of x times the natural logarithm of b. This relationship holds for any base b and exponent x and shows the connection between these two mathematical concepts.

Step by step solution

01

Write the definition of the natural logarithm

The natural logarithm is the inverse function of the exponential function with base e. That is, \(\ln(x)\) is the power to which we need to raise the number e (Euler's number, approximately equal to 2.718) to get x. Mathematically, we can write this as: $$\ln(x) = y \iff e^{y} = x$$
02

Find the natural logarithm of both sides

To make use of the natural logarithm, we can find the natural logarithm of both sides of the equation \(b^x = e^{x \ln b}\). $$\ln(b^x) = \ln(e^{x \ln b})$$
03

Apply the logarithm power rule

Using the power rule of logarithms, which states \(\ln(a^b) = b \ln(a)\), we can simplify both sides of the equation: $$x \ln(b) = x \ln b$$
04

Conclusion

Since \(\ln(b^x) = x \ln b\) and keeping in mind that raising to the power of a logarithm is equal to the logarithm of a power, we can conclude that \(b^x = e^{x \ln b}\). This relationship holds for any base b and exponent x, and demonstrates the connection between the exponential function and the natural logarithm function.

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Key Concepts

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

Natural Logarithm
Understanding the natural logarithm is key to grasping the relationship between exponential and logarithmic functions. The natural logarithm, denoted as \( \ln(x) \), is essentially the power we need to raise Euler's number \( e \) to obtain \( x \).
This is written as \( \ln(x) = y \) if and only if \( e^y = x \). Here, \( e \) is a constant approximately equal to 2.718, representing the base of the natural logarithm.
  • The notation \( \ln(x) \) implies that we are dealing with a logarithm whose base is \( e \).
  • It鈥檚 essential because many natural processes grow exponentially at base \( e \), making it highly applicable in real-world scenarios like compound interest and population growth.
In the context of the equation \( b^x = e^{x \ln b} \), we see the utility of natural logarithms in expressing powers and simplifying expressions where exponential terms are involved.
Inverse Functions
Inverse functions allow us to reverse the effect of a given function. In the case of exponential functions and logarithms, they are inverses of each other.
This means that if you perform an exponential operation on a number and then take the logarithm, you'll get back the original number you started with.
  • For exponential functions, the inverse is the logarithm function.
  • The natural logarithm \( \ln(x) \) inverts the exponential function \( e^x \).
This inverse relationship is what makes it possible to transform equations and expressions, such as \( b^x = e^{x \ln b} \), into more manageable forms for solving problems in mathematics and science.
Whenever you encounter exponential equations, considering their inverses can often lead to simpler and more elegant solutions.
Power Rule of Logarithms
The power rule of logarithms is a pivotal tool for simplifying expressions involving logarithms.
This rule states that \( \ln(a^b) = b \ln(a) \), which means when we raise a number to a power inside a logarithm, we can bring the exponent out in front as a coefficient.
  • This property is useful because it turns multiplication into addition, simplifying complex equations.
  • Applying this rule helps solve for unknowns in exponential equations by making them linear.
In our example, using the power rule on \( \ln(b^x) \) simplifies it to \( x \ln(b) \).
This simplification allows us to compare and equate expressions more directly, as seen in the equation \( b^x = e^{x \ln b} \), illustrating the consistent property of powers and logarithms in mathematics.

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Most popular questions from this chapter

Let \(f\) and \(g\) be differentiable functions with \(h(x)=f(g(x)) .\) For a given constant \(a,\) let \(u=g(a)\) and \(v=g(x),\) and define $$H(v)=\left\\{\begin{array}{ll} \frac{f(v)-f(u)}{v-u}-f^{\prime}(u) & \text { if } v \neq u \\ 0 & \text { if } v=u \end{array}\right.$$ a. Show that \(\lim _{y \rightarrow u} H(v)=0\) b. For any value of \(u,\) show that $$f(v)-f(u)=\left(H(v)+f^{\prime}(u)\right)(v-u)$$ c. Show that $$h^{\prime}(a)=\lim _{x \rightarrow a}\left(\left(H(g(x))+f^{\prime}(g(a))\right) \cdot \frac{g(x)-g(a)}{x-a}\right)$$ d. Show that \(h^{\prime}(a)=f^{\prime}(g(a)) g^{\prime}(a)\)

Let \(f(x)=x e^{2 x}\) a. Find the values of \(x\) for which the slope of the curve \(y=f(x)\) is 0 b. Explain the meaning of your answer to part (a) in terms of the graph of \(f\)

a. Determine the points where the curve \(x+y^{3}-y=1\) has a vertical tangent line (see Exercise 60 ). b. Does the curve have any horizontal tangent lines? Explain.

A \(\$ 200\) investment in a savings account grows according to \(A(t)=200 e^{0.0398 t}\), for \(t \geq 0,\) where \(t\) is measured in years. a. Find the balance of the account after 10 years. b. How fast is the account growing (in dollars/year) at \(t=10 ?\) c. Use your answers to parts (a) and (b) to write the equation of the line tangent to the curve \(A=200 e^{0.0398 t}\) at the point \((10, A(10))\)

Use implicit differentiation to find\(\frac{d y}{d x}.\) $$\cos y^{2}+x=e^{y}$$
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