Problem 35 - Algebra and Trigonometry

Question

Solve the logarithmic equation for \(x\) $$ \ln x=10 $$

Step-by-Step Solution

Answer

The solution for \( x \) is \( e^{10} \).

Step 1: Understanding the natural logarithm

The equation given is \( \ln x = 10 \). Here, \( \ln \) stands for the natural logarithm, which is the logarithm with base \( e \). The equation asks us to find the value of \( x \) such that its natural logarithm equals 10.

Step 2: Recall the inverse property of logarithms

The inverse property of logarithms states that if \( \ln a = b \), then \( a = e^b \). This means the exponential function with base \( e \) is the inverse of the natural logarithm.

Step 3: Apply the inverse property

Using the property from Step 2, since \( \ln x = 10 \), we can write \( x = e^{10} \). This gives us the expression for \( x \) in terms of the exponential function with base \( e \).

Step 4: Interpret the result

The solution, \( x = e^{10} \), represents the value of \( x \) such that the natural logarithm equals 10. This is the unique solution as the natural logarithm is a one-to-one function.

Key Concepts

Natural Logarithm

A natural logarithm, denoted as \( \ln \), is a special logarithm that uses the base \( e \). The number \( e \) is an irrational and transcendental number approximately equal to 2.71828. Natural logarithms are used frequently in science and engineering because they relate exponential growth processes naturally occurring in the physical world.

In the equation \( \ln x = 10 \), the natural logarithm tells us that we are looking for a number \( x \) such that, when the exponential function with base \( e \) is applied, we obtain that number which has a natural logarithm of 10. This means \( x \) will be a large number because log values represent exponents. The natural logarithm is particularly useful because it's easier to understand and manipulate compared to other logarithmic bases, especially when dealing with exponential growth, decay, and compound interest.

• The base of the natural logarithm is \( e \).
• \( \ln \) stands for the natural logarithm.
• It helps in solving problems involving exponential growth and decay.

Exponential Function

An exponential function is a mathematical function of the form \( f(x) = a^x \), where \( a \) is a constant known as the base and \( x \) is the exponent. When we refer to the natural exponential function, \( e \) is used as the base. Therefore, the natural exponential function can be written as \( f(x) = e^x \).

This function is pivotal in various fields because it describes processes where growth acceleration is proportional to current value, such as population growth, radioactive decay, and continuously compounded interest.

• The exponential function with base \( e \) is written as \( e^x \).
• It models natural processes where change accelerates.
• In solving \( \ln x = 10 \), we find \( x \) using \( x = e^{10} \).

Inverse Property of Logarithms

Logarithms and exponential functions are inverse operations, meaning they undo each other. The inverse property of logarithms states that if you take the logarithm of a number and then exponentiate it, you return to the original number. In mathematical terms, if \( \ln a = b \), then \( a = e^b \).

This property is very useful because it provides a way to solve equations involving logarithms. If we have \( \ln x = 10 \), we can use this inverse property to find \( x \), which leads us to \( x = e^{10} \). This transformation flips the natural logarithmic statement into an exponential one, granting us the solution.

• The inverse function of \( \ln x \) is \( e^x \).
• It simplifies solving logarithmic equations.
• Using \( \ln x \), we find \( x = e^{10} \), utilizing the inverse property.