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Chapter 5: Problem 50

Convert to an exponential equation. \(\ln 0.38=-0.9676\)

Short Answer

Expert verified
The exponential form is \( e^{-0.9676} = 0.38 \).

Step by step solution

01

Understand the Natural Logarithm

The natural logarithm, \(\text{ln}\), is the logarithm to the base \(e\), where \(e\) is approximately equal to 2.71828. In this exercise, we have \( \text{ln} (0.38) = -0.9676 \).
02

Rewrite the Logarithmic Equation as an Exponential Equation

To convert a natural logarithm equation to its exponential form, use the property \( \text{ln}(a) = b \) implies \( e^b = a \). For the given equation \( \text{ln}(0.38) = -0.9676 \), rewrite it as \( e^{-0.9676} = 0.38 \).
03

Summarize the Exponential Equation

The equivalent exponential form of the logarithmic equation \( \text{ln}(0.38) = -0.9676 \) is \( e^{-0.9676} = 0.38 \).

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Key Concepts

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

natural logarithms
The natural logarithm, denoted as \(\text{ln}\), is a logarithm with the base \(e\), which is an irrational number approximately equal to 2.71828. Natural logarithms are used frequently in mathematics, especially in calculus and complex calculations involving exponential growth and decay. One of the main properties of the natural logarithm is that it undoes the action of the exponential function with base \(e\). For example, if you have \(\text{ln}(x) = y\), you can rewrite this in its exponential form as \(e^y = x\). This relationship is critical when converting between logarithmic and exponential forms.
logarithmic equations
Logarithmic equations involve the logarithm of a variable or number. They often need to be solved by converting them into an equivalent exponential equation. For example, the equation \(\text{ln}(0.38) = -0.9676\) needs to be rewritten in its exponential form for easier interpretation and solution. To do this, we use the property that \(\text{ln}(a) = b\) implies \(e^b = a\). Applying this to our example, \(\text{ln}(0.38) = -0.9676\) can be rewritten as the exponential equation \(e^{-0.9676} = 0.38\). Understanding this transformation is crucial for solving logarithmic equations.
base e
The number \(e\) is one of the most important constants in mathematics, approximately equal to 2.71828. It serves as the base for natural logarithms and has unique properties that make it incredibly useful in various fields, including calculus, complex analysis, and mathematical modeling of growth processes. The function \(e^x\) is its own derivative and integral, which is why \(e\) is vital in differential equations and exponential growth models. When converting between logarithmic and exponential forms, recognizing the role of base \(e\) helps simplify and solve equations efficiently. For instance, starting with \(\text{ln}(0.38) = -0.9676\), understanding that \(e\) serves as the base allows you to easily rewrite this as \(e^{-0.9676} = 0.38\). This showcases the interplay between the natural logarithm and the exponential function with base \(e\).

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Most popular questions from this chapter

Use a graphing calculator to find the approximate solutions of the equation. $$\log _{5}(x+7)-\log _{5}(2 x-3)=1$$

Determine whether each of the following is true or false. Assume that \(a, x, M,\) and \(N\) are positive. $$\log _{a} M-\log _{a} N=\log _{a} \frac{M}{N}$$

Compound Interest. Suppose that \(\$ 750\) is invested at \(7 \%\) interest, compounded semiannually. a) Find the function for the amount to which the investment grows after \(t\) years. b) Graph the function. c) Find the amount of money in the account at \(t=1,6\) \(10,15,\) and 25 years. d) When will the amount of money in the account reach \(\$ 3000 ?\)

Consider quadratic functions ( \(a\) )-( h ) that follow. Without graphing them, answer the questions below. a) \(f(x)=2 x^{2}\) b) \(f(x)=-x^{2}\) c) \(f(x)=\frac{1}{4} x^{2}\) d) \(f(x)=-5 x^{2}+3\) e) \(f(x)=\frac{2}{3}(x-1)^{2}-3\) f) \(f(x)=-2(x+3)^{2}+1\) g) \(f(x)=(x-3)^{2}+1\) h) \(f(x)=-4(x+1)^{2}-3\) Which graph has vertex \((-3,1) ?\)

E-Cigarette SE-Cigarette Sales. The electronic cigarette was launched in 2007 , and since then sales have increased from about \(\$ 20\) million in 2008 to about \(\$ 500\) millionales. The electronic cigarette was launched in 2007 , and since then sales have increased from about \(\$ 20\) million in 2008 to about \(\$ 500\) million in 2012 (Sources: UBS; forbes, \(\mathrm{com}\) ). The exponential function $$ S(x)=20.913(2.236)^{x} $$ where \(x\) is the number of years after \(2008,\) models the sales, in millions of dollars. Use this function to estimate the sales of e-cigarettes in 2011 and in 2015 . Round to the nearest million dollars.
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