Problem 12 - Elementary and Intermediate Algebra

Question

Fill in the blanks. The logarithmic equation \(\ln x=1.5318\) is equivalent to the exponential equation __\(=\)___.

Step-by-Step Solution

Answer

The exponential equation is \(e^{1.5318} = x\).

Step 1: Understanding Logarithmic Form

The equation given is in logarithmic form. This form, \( \ln x = 1.5318 \), represents a logarithm with base \( e \) where \( x \) is the number we are finding the log of.

Step 2: Converting to Exponential Form

To convert \( \ln x = 1.5318 \) from logarithmic to exponential form, recall that \( \ln x = \log_e x = y \) implies \( e^y = x \). Therefore, our equation implies \( e^{1.5318} = x \).

Key Concepts

Exponential Equations

Exponential equations are a key concept in mathematics where the unknown variable appears in the exponent. These equations often look like \[ a^x = b \]where \(a\) is a constant base, \(x\) is the exponent, and \(b\) is another constant. Solving exponential equations usually involves finding the value of the variable that makes the equation true.

• One major way to solve them is by isolating the exponential expression and then taking logarithms on both sides.
• Just like how addition and subtraction or multiplication and division are inverse operations, exponents and logarithms are also inverse operations.
• Thus, if you have an equation like \(e^{x} = c\), you can take the natural logarithm of both sides to find \(x\).

Exponential equations are essential in fields like science and finance, where growth and decay processes follow this form.

Natural Logarithm

The natural logarithm, often denoted as \(\ln\), is a special type of logarithm with the base of the constant \(e\), approximately equal to 2.71828. It satisfies the property:\[\ln(e^x) = x.\]

• The natural logarithm is commonly used in calculus and other areas of mathematics because it simplifies calculations with continuous growth or decay.
• It's particularly important because it relates the rate of growth of a function to the function itself.
• When you see \(\ln(x)\), think of it as the power you have to raise \(e\) to in order to get \(x\).

Understanding natural logarithms is vital for working with exponential equations, as they allow us to convert these tricky expressions into simpler forms.

Converting Equations

Converting equations between logarithmic and exponential forms is critical for understanding and solving many mathematical problems. This process often simplifies the equation and reveals the value of the unknown variable.
To convert a logarithmic equation into its exponential form, use the relationship:

• If \(\ln(x) = y\) (a logarithm with base \(e\)), it is equivalent to \(e^y = x\).
• This conversion allows you to solve for \(x\) by exponentiating both sides.
• For example, from \(\ln(x) = 1.5318\), we deduce that \(e^{1.5318} = x\), allowing easy computation of \(x\).

Being comfortable with switching between these formats is crucial for calculus, algebra, and various applications in physics and engineering. It often helps to focus on understanding the roles logarithms and exponents play in these equations.