Problem 19 - Contemporary Precalculus

Question

Translate the given exponential statement into an equivalent logarithmic statement. $$e^{3.25}=25.79$$

Step-by-Step Solution

Answer

Question: Convert the exponential equation \(e^{3.25} = 25.79\) into a logarithmic equation. Answer: The equivalent logarithmic equation for the given exponential equation is \(\ln{25.79} = 3.25\).

Step 1: Identify the exponential equation

The given exponential equation is: $$e^{3.25} = 25.79$$

Step 2: Rearrange as a logarithmic equation

We can rearrange the equation as a logarithmic equation with the base e (natural logarithm, written as \(\ln\)). The general form of a logarithmic equation is: $$\log_{a}{b} = x$$ In our case, \(a\) is the base (e), \(b\) is the result (25.79), and \(x\) is the exponent (3.25).

Step 3: Rewrite as a logarithmic equation

Using the relationship between exponential and logarithmic equations, we can rewrite the given equation as: $$\ln{25.79} = 3.25$$ This is the equivalent logarithmic statement for the given exponential statement: \(e^{3.25} = 25.79\).

Key Concepts

Natural Logarithm

The natural logarithm is a special type of logarithm that has the mathematical constant e (approximately 2.71828) as its base. It's written as \(\ln(x)\) and is the inverse function of the exponential function \(e^x\). In simpler terms, the natural logarithm of a number answers the question: 'To what power must e be raised, to get this number?’

For example, if we have the equation \(e^y = x\), then the natural logarithm of \(x\) is \(y\), expressed as \(\ln(x) = y\). This relationship is crucial because it allows us to solve exponential equations involving e by transforming them into logarithmic equations, which are often easier to work with.

Exponential Equations

Exponential equations feature variables in the exponent and can often be recognized by their base raised to a power, like \(a^x = b\). These equations can describe phenomena that exhibit exponential growth or decay, such as population growth, radioactive decay, or compound interest.

One way to solve these equations is to use logarithms. The base of the logarithm used to solve the equation typically matches the base of the exponent. For bases other than e, this means using the logarithm function \(\log_b(x)\), but for the natural base e, the natural logarithm \(\ln(x)\) is more appropriate. Harnessing the power of logarithms allows us to 'bring down' the exponent, making it possible to solve for the variable.

Logarithmic Equations

Logarithmic equations contain logarithms and usually present a relationship where the logarithm of a number is set equal to a value. The structure of such an equation is \(\log_a(b) = c\), where \(a\) is the base of the logarithm, \(b\) is the argument, and \(c\) is the value the logarithm equals to.

When solving logarithmic equations, the goal is to isolate the logarithm on one side and convert it to its equivalent exponential form to solve for the argument. It's crucial to remember that a logarithm basically asks the question, 'To what power must the base be raised to produce this argument?' This conceptual understanding can significantly aid in the transformation and solution of logarithmic equations.

Logarithmic Form

The logarithmic form of an expression converts an exponential equation into its logarithmic counterpart. The conversion process from exponential form \(a^x = b\) to logarithmic form is \(\log_a(b) = x\). This transformation is valuable because it enables us to solve for variables that were once exponents, which is difficult with basic algebraic tools.

As shown in the original exercise, \(e^{3.25} = 25.79\), by converting to logarithmic form we get \(\ln(25.79) = 3.25\). By understanding and applying the logarithmic form, students can navigate through problems involving exponential functions with more ease and confidence.