Problem 14 - Algebra and Trigonometry

Question

\(9-14\) Express the equation in logarithmic form. $$ \begin{array}{lll}{\text { (a) } e^{x+1}=0.5} & {\text { (b) } e^{0.5 x}=t}\end{array} $$

Step-by-Step Solution

Answer

(a) In logarithmic form: \(\ln(0.5) = x + 1\); (b) \(\ln(t) = 0.5x\).

Step 1: Understanding the Task

We need to convert two exponential equations into their equivalent logarithmic forms. The general form for converting an exponential equation, such as \(a^b = c\), to a logarithmic equation is \(\log_a{c} = b\). Here, \(a\) is the base of the exponential equation.

Step 2: Identify the Base for Equation (a)

In equation (a), \(e^{x+1} = 0.5\), the base is \(e\), which is the base of the natural logarithm.

Step 3: Convert Equation (a) to Logarithmic Form

Using the exponential to logarithmic conversion rule, the equation \(e^{x+1} = 0.5\) can be written in logarithmic form as \(\ln(0.5) = x + 1\), where \(\ln\) denotes the natural logarithm.

Step 4: Simplify Logarithmic Form of Equation (a)

Rearrange the logarithmic form to isolate \(x\): \(x = \ln(0.5) - 1\).

Step 5: Identify the Base for Equation (b)

In equation (b), \(e^{0.5x} = t\), again, the base is \(e\).

Step 6: Convert Equation (b) to Logarithmic Form

Using the conversion rule, the equation \(e^{0.5x} = t\) is rewritten as \(\ln(t) = 0.5x\).

Step 7: Simplify Logarithmic Form of Equation (b)

To express \(x\), rearrange the equation: \(x = \frac{2}{1}\ln(t)\), since \(0.5x\) implies dividing both sides by 0.5 or multiplying by 2.

Key Concepts

Exponential Equations

Exponential equations are those where a variable appears in the exponent. These equations typically have the form \(a^b = c\), where \(a\) is the base, \(b\) is the exponent, and \(c\) is a constant or expression. Solving these equations often involves finding the value of the exponent that makes the equation true.

Here are some important features of exponential equations:

• The base is a constant number. In many cases, it is the number \(e\) (approximately 2.718), especially in natural exponential equations, making them involve the natural logarithm.
• The exponent contains the variable we are solving for, which makes these equations different from polynomial equations.
• Exponential equations model processes with constant relative growth or decay, such as population growth or radioactive decay.

To solve these equations, we often convert them into a logarithmic form using specific rules. This simplification allows us to find the unknown variable easier.

Natural Logarithm

The natural logarithm, denoted by \( \ln \), is a special type of logarithm with the base \(e\). In other words, \[\ln(x) = \log_e(x)\] It is widely used in various fields like mathematics, physics, and engineering because of its natural appearance in mathematical analysis.

Some key points about the natural logarithm include:

• It is the inverse operation of taking an exponential with base \(e\). This means that solving the equation \(e^x = y\) for \(x\) involves taking the natural logarithm of \(y\), becoming \(x = \ln(y)\).
• The natural logarithm has properties similar to common logarithms (e.g., \(\ln(ab) = \ln(a) + \ln(b)\), \(\ln(a^b) = b \cdot \ln(a)\), etc.).
• It simplifies solving exponential equations, making it a crucial tool when working with equations involving \(e\).

Understanding the natural logarithm is essential for converting exponential equations into a more solvable form.

Converting Exponential to Logarithmic

Converting an exponential equation to a logarithmic form is a common method used to solve for unknown variables that appear as exponents. The conversion uses the key formula:\[ b = \log_a(c) \Rightarrow a^b = c \]This equation expresses that the logarithm of \(c\) to the base \(a\) equals \(b\).

Here's how to use this conversion:

• Identify the base \(a\) in the exponential equation. For natural exponential equations, this base is the constant \(e\).
• Convert \(a^b = c\) to \(b = \log_a(c)\). If \(a = e\), then use the natural logarithm, giving \(b = \ln(c)\).
• Once in logarithmic form, simplify to isolate the variable and make solving straightforward.

In solving the original exercise problems, we used this process to turn \(e^{x+1} = 0.5\) into \(x+1 = \ln(0.5)\), and \(e^{0.5x} = t\) into \(0.5x = \ln(t)\). This conversion made it easier to subsequently solve for \(x\).