Problem 40
Question
Use your calculator to find \(x\) when given \(\ln x\). Express answers to five significant digits. $$ \ln x=-2.3745 $$
Step-by-Step Solution
Verified Answer
\( x \approx 0.093247 \)
1Step 1: Understanding the Problem
The natural logarithm (ln) of a number is given, and we need to find the original number (x). The equation provided is \( \ln x = -2.3745 \). Our goal is to solve for \( x \).
2Step 2: Recall the Inverse Operation
Remember that the inverse operation of the natural logarithm (ln) is the exponential function, denoted as \( e \). Hence, to find \( x \) from \( \ln x \), you use the formula \( x = e^{\ln x} \).
3Step 3: Rewrite the Equation
Rewrite the equation using the inverse operation: \( x = e^{-2.3745} \). This expression tells us we need to raise the value of the mathematical constant \( e \) (approximately 2.71828) to the power of -2.3745.
4Step 4: Use a Calculator
Use a scientific calculator to compute \( x = e^{-2.3745} \). Make sure your calculator is set to at least 5 significant figures to get the precise value.
5Step 5: Find the Result
After computing the expression, you will find that \( x \approx 0.093247 \). Therefore, \( x \) is approximately 0.093247 to five significant digits.
Key Concepts
Natural LogarithmExponential FunctionSignificant Digits
Natural Logarithm
The natural logarithm, often abbreviated as "ln," is a mathematical function that gives the time needed for a quantity to reach a certain level of growth, assuming a continuous rate of growth. The base of a natural logarithm is the number \( e \), which is approximately equal to 2.71828. This makes the natural logarithm the inverse process of the exponential function with base \( e \).
In simpler terms, when you see something like \( \ln(x) \), it means: "To what power do we need to raise \( e \) to get \( x \)?" For instance, \( \ln(e) = 1 \) because \( e^1 = e \). Remembering this can help you navigate through problems involving logarithms and the need to "undo" such expressions is common when solving equations.
In simpler terms, when you see something like \( \ln(x) \), it means: "To what power do we need to raise \( e \) to get \( x \)?" For instance, \( \ln(e) = 1 \) because \( e^1 = e \). Remembering this can help you navigate through problems involving logarithms and the need to "undo" such expressions is common when solving equations.
Exponential Function
An exponential function is one of the most common types of mathematical functions and is expressed as \( e^x \) in its natural form. The symbol \( e \) represents Euler's number, an irrational number approximately equal to 2.71828. When using exponential functions, the exponent \( x \) can be any real number, positive or negative.
The key to understanding exponential functions is realizing how they relate to their inverse — the natural logarithm. Whenever you take the natural logarithm of a number, applying the exponential function with base \( e \) allows you to reverse the process. For instance, in the given exercise, the process of finding \( x \) when \( \ln(x) = -2.3745 \) involves using an exponential calculation to recover the original number that was "logged." This is done by calculating \( e^{-2.3745} \).
Exponential functions are vital across many fields, including finance, physics, and biology because they model continuous growth or decay effectively.
The key to understanding exponential functions is realizing how they relate to their inverse — the natural logarithm. Whenever you take the natural logarithm of a number, applying the exponential function with base \( e \) allows you to reverse the process. For instance, in the given exercise, the process of finding \( x \) when \( \ln(x) = -2.3745 \) involves using an exponential calculation to recover the original number that was "logged." This is done by calculating \( e^{-2.3745} \).
Exponential functions are vital across many fields, including finance, physics, and biology because they model continuous growth or decay effectively.
Significant Digits
Significant digits, also known as significant figures, are a way of expressing precision in a numerical value. Significant digits can tell you how exact a measurement is. In calculations, keeping track of significant digits helps prevent overestimating the precision of your results.
When expressing a number in terms of significant digits, each non-zero digit is significant, any zero between non-zero digits is also significant, and any trailing zero in the decimal portion is significant if it serves to indicate precision. For instance, in the result \( x \approx 0.093247 \), each digit is considered significant, reflecting the precision of the computation.
In scientific calculations like the one performed, it's important to ensure that the outcome — in this case \( 0.093247 \) — aligns with the problem's requirement for precision by getting exactly five significant digits. This ensures consistency and accuracy in the reported results.
When expressing a number in terms of significant digits, each non-zero digit is significant, any zero between non-zero digits is also significant, and any trailing zero in the decimal portion is significant if it serves to indicate precision. For instance, in the result \( x \approx 0.093247 \), each digit is considered significant, reflecting the precision of the computation.
In scientific calculations like the one performed, it's important to ensure that the outcome — in this case \( 0.093247 \) — aligns with the problem's requirement for precision by getting exactly five significant digits. This ensures consistency and accuracy in the reported results.
Other exercises in this chapter
Problem 39
Graph each of the exponential functions. $$ f(x)=2^{-x-2} $$
View solution Problem 40
Approximate each logarithm to three decimal places. $$ \log _{5} 0.047 $$
View solution Problem 40
Evaluate each logarithmic expression. $$ \log _{2}\left(\log _{5} 5\right) $$
View solution Problem 40
(a) find \(f^{-1}\) and (b) verify that \(\left(f \circ f^{-1}\right)(x)=x\) and \(\left(f^{-1} \circ f\right)(x)=x\). $$ f(x)=-5 x+6 $$
View solution