Problem 32 - Algebra 2 and Trigonometry

Question

In \(21-32,\) for each given logarithm, find \(x,\) the antilogarithm. Write the answer to four decimal places. $$ \ln x=-2 $$

Step-by-Step Solution

Answer

The antilogarithm \( x \) is approximately 0.1353.

Step 1: Understand the Given Equation

We are given the equation \( \ln x = -2 \). This is a natural logarithm, which means it uses base \( e \), where \( e \approx 2.71828 \).

Step 2: Convert to Exponential Form

To find the antilogarithm, we need to convert the logarithmic equation to its exponential form. The formula relating logarithms and exponents is \( \ln x = -2 \) can be rewritten as \( x = e^{-2} \).

Step 3: Calculate the Exponential Value

Now, we calculate \( e^{-2} \). The value of \( e \) is approximately \( 2.71828 \). Therefore, \( e^{-2} = \frac{1}{e^2} \approx \frac{1}{2.71828^2} \approx 0.1353 \).

Step 4: Round to Four Decimal Places

The calculated value of \( x \) is approximately \( 0.1353352832 \). Rounding this to four decimal places gives \( 0.1353 \).

Key Concepts

Understanding Natural Logarithms

The natural logarithm, often noted as "ln," is a specific type of logarithm that uses the mathematical constant \( e \) as its base. The number \( e \) is an irrational number approximately equal to 2.71828. Natural logarithms are frequently used in calculus and complex mathematical problems because they have properties that make differentiation and integration simpler.

• When you see an expression like \( \ln(x) = y \), it tells you that \( x \) is \( e \) raised to the power of \( y \).
• The inverse of a natural logarithm is an exponential function, which allows us to "undo" the logarithm operation.

In the given problem, if \( \ln(x) = -2 \), we are seeking the value of \( x \) such that when you take the natural logarithm of \( x \), you get -2.By converting from a logarithmic form to an exponential form, we can solve for \( x \).

Converting Logarithmic Equations to Exponential Form

Converting a logarithmic equation to its exponential form is a crucial step to find the value of \( x \) or the antilogarithm. Exponential functions allow us to work with equations in a more straightforward way when logarithms are involved.The general rule is: if you have a logarithmic equation \( \log_b(a) = c \), it can be expressed in exponential form as \( a = b^c \).

• For natural logarithms, \( \ln(x) = c \) becomes \( x = e^c \).
• This step is really about translating the problem into a form where \( x \) can be directly calculated.

In the exercise, \( \ln x = -2 \) becomes \( x = e^{-2} \). This transformation makes it possible to evaluate \( x \) directly since \( e^{-2} \) simply requires us to perform exponential operations with the number \( e \).

Rounding to Decimal Places

Once you have calculated a number, you often need to round it to a certain number of decimal places, especially in exercises that specify an exactness requirement, like rounding to four decimal places. Rounding helps present many decimal places in a more manageable and readable form.Rounding is performed by looking at the number immediately to the right of the last decimal place you want:

• If this number is 5 or greater, you round up.
• If it's less than 5, you round down, or simply leave the number as it is.

In our exercise, after calculating \( e^{-2} \), we arrived at the value of approximately 0.1353352832.

• Looking at the fifth digit, which is 5, we round the fourth place digit from 3 up to 4.
• The final rounded answer is 0.1353 when rounding to four decimal places.

Rounding ensures your answers meet the precision needed without unnecessary complexity.