Problem 13 - Algebra and Trigonometry

Question

\(9-14\) Express the equation in logarithmic form. $$ \begin{array}{ll}{\text { (a) } e^{x}=2} & {\text { (b) } e^{3}=y}\end{array} $$

Step-by-Step Solution

Answer

(a) \(x = \ln(2)\), (b) \(3 = \ln(y)\).

Step 1: Convert to Logarithmic Form (Part a)

Start by identifying the exponential equation, which is \(e^x = 2\). The logarithmic form is derived from the definition of the natural logarithm, which is the logarithm to the base \(e\). Thus, the equation \(e^x = 2\) can be written in logarithmic form as \(x = \ln(2)\).

Step 2: Convert to Logarithmic Form (Part b)

For the second equation \(e^3 = y\), follow the same logic. The logarithmic form of this equation involves identifying the base \(e\), which leads to \(3 = \ln(y)\).

Key Concepts

Natural Logarithm

The natural logarithm is a special kind of logarithm that has the constant e as its base. The number e is a mathematical constant approximately equal to 2.71828. It's crucial in the fields of calculus and complex analysis. In simpler terms, you can think of the natural logarithm as the "inverse" of the exponential function with base e.
The notation for the natural logarithm is \( \ln \), and it serves a specific purpose:

• When you want to know what exponent you need to raise e to, so it equals some other number, you use the natural logarithm.
• For example, if you want to solve \( e^x = 2 \), you use \( x = \ln(2) \).

Understanding natural logarithms helps in solving exponential equations, where the base is e. They appear frequently in problems involving continuous growth or decay, such as population growth, radioactive decay, and interest calculations.

Exponential Equations

Exponential equations are mathematical expressions where the variable appears in the exponent. They often express rapid growth or decay and play a significant role in scientific contexts. The general form of an exponential equation is \( a^x = b \), where a is a positive number, not equal to 1, and b is a positive constant.

• Exponential equations can be solved by comparing the bases on both sides or using logarithms to bring the variable down from the exponent.
• In the exercise, you encountered \( e^x = 2 \). To solve it, you convert it into a logarithmic equation, helping to isolate the variable x.

Exponential growth occurs when the rate of change of a quantity is proportional to its current value, such as with population or investment growth. Conversely, exponential decay describes processes like radioactive decay or depreciation. Knowing how to manipulate exponential forms is key to mastering such problems.

Logarithms

Logarithms are the inverse operation of exponentiation, essentially answering the question, "To what power must a given base be raised, to produce a certain number?" They make it easier to work with large numbers by converting multiplication into addition, which is a foundational concept in algebra.
A logarithmic function is expressed as \( \log_b(x) = y \), meaning the base b raised to the power y equals x (i.e., \( b^y = x \)).

• In the context of the exercise, you used the natural logarithm, \( \ln \), to express exponential equations in logarithmic form.
• Logs provide an efficient tool for solving equations where the variable is an exponent.

By understanding how logarithms transform relationships in mathematics, you can solve complex equations with more ease. It simplifies calculations involving exponential growth or decay, making it an essential concept in math and science.