Problem 25 - College Algebra

Question

For the following exercises, rewrite each equation in logarithmic form. $$e^{k}=h$$

Step-by-Step Solution

Answer

\( \ln(h) = k \)

Step 1: Understand Exponential to Logarithmic Form

The exponential equation given is \( e^k = h \). In exponential equations, \( e \) represents the base, \( k \) is the exponent, and the result is \( h \). To convert this into a logarithmic form, we need to switch from showing the power to showing how we reach the power using logarithms.

Step 2: Apply Logarithmic Definition

Recall that an equation in exponential form \( a^b = c \) can be rewritten as a logarithm \( \log_a(c) = b \). Here, \( a \) is the base, \( b \) is the solution to the exponent (what the logarithmic form solves for), and \( c \) is the result. For the equation \( e^k = h \), we apply the same principle.

Step 3: Rewrite in Logarithmic Form

Using the definition, rewrite \( e^k = h \) in logarithmic form: This becomes \( \log_e(h) = k \). In this form, \( \log_e \) represents a natural logarithm, often written as \( \ln \). Therefore, it simplifies to \( \ln(h) = k \).

Key Concepts

Exponential Equations

Exponential equations are mathematical expressions where a constant base is raised to a variable exponent to produce a particular outcome. A common form of these equations is written as \( a^x = y \), where \( a \) is the base, \( x \) is the exponent, and \( y \) is the result. Exponential equations allow us to express large numbers succinctly or model real-world phenomena that change rapidly, such as population growth or radioactive decay. These equations are powerful because they describe the means of exponential growth or decay—both critical concepts in fields like finance, biology, and physics. When solving exponential equations, we often need to isolate the exponent, and this is where understanding their conversion to logarithmic form becomes beneficial. It helps us deal with the exponent directly and solve for unknowns that aren't easily tackled when using exponents alone.

Natural Logarithm

The natural logarithm is a specific logarithm with the irrational number \( e \) (approximately 2.718) as its base. It is denoted as \( \ln \), so \( \ln(x) \) is essentially the same as \( \log_e(x) \). Natural logarithms are especially important in calculus and higher mathematics because of their inherent simplicity when differentiating or integrating functions that involve exponential growth or decay. They also appear frequently in complex mathematical models representing natural phenomena. Here are some important properties of natural logarithms:

• \( \ln(1) = 0 \), since \( e^0 = 1 \).
• \( \ln(e) = 1 \), because \( e^1 = e \).
• \( \ln(xy) = \ln(x) + \ln(y) \).
• \( \ln\left(\frac{x}{y}\right) = \ln(x) - \ln(y) \).
• \( \ln(x^a) = a \cdot \ln(x) \).

These properties help simplify complex calculations and solve exponential equations efficiently by reducing them into simpler, additive forms.

Conversion of Exponential to Logarithmic Form

The process of converting an exponential equation to its logarithmic form is essential for solving equations where the variable is in the exponent. This conversion provides a way to 'bring down' the exponent, offering a clearer path to isolate and solve for variables. Consider the general equation \( a^b = c \), where \( a \) is the base, \( b \) is the exponent, and \( c \) is the result. To convert this equation to logarithmic form, it takes the form \( \log_a(c) = b \). This tells us that the exponent \( b \) is the power to which the base \( a \) must be raised to yield \( c \). Let's apply this to our specific example: given the equation \( e^k = h \), we want to convert this to a logarithmic form. According to the conversion principle, it becomes \( \log_e(h) = k \). Since \( \log_e \) is the natural logarithm written as \( \ln \), it simplifies to \( \ln(h) = k \).This conversion is particularly useful because it transforms exponential equations into forms that are easier to manipulate. This is especially handy in calculus and other fields of mathematics where solving for a variable exponent directly can be quite complex.